Evridiki Papastavrou

Andreas Charalambous

Georgios Efstathiou

Keywords: behavio(u)rs, caring, literature review, nurse, patient, perception, quantitative systematic review

2. The key words included were changed to match the Journal's guidelines.

These key words were included: behavio(u)rs, caring, literature review, nurse, patient, perception, quantitative systematic review 3. The paper was proof read and corrected by the Editage.

4. A web-file was created to include the "characteristics of the reviewed studies" 5. Under the review "design" heading, the following sentence was included: "For this quantitative comparative review, a narrative summary approach was adopted for reporting the findings".

A web-file was created to include Table 1 -Quality Appraisal

7. An additional web-file was included to list the excluded studies 8. The title was changed according to the reviewers' comments as follows:

Nurses and patients perceptions of caring behaviours: Quantitative systematic review of comparative studies 9. The revised paper's word count is 4475. as the 'art' of nursing which is conveyed as the exercise of a human skill, the skill of caring [START_REF] Clifford | Caring: fitting the concept to nursing practice[END_REF][START_REF]Art, Science, or Both? Keeping the Care in Nursing[END_REF]. In the current global climate in nursing (and health care in general), the concept of 'caring' needs to be re-contextualized in order to reflect current developments and changes in the way nursing is practiced by nurses and perceived by patients. The emphasis on intervention outcomes in health care creates a need to develop a clear understanding of which nursing behaviours convey caring, to explain patient outcomes from nursing practice, and to predict patient well-being and health.

Caring presents itself as a nebulous concept in nursing, one that has triggered over the years intense and constant efforts to capture its meaning and particularly its meaning as it is manifested in the nursing profession [START_REF] Sherwood | Meta-synthesis of qualitative analyses of caring: defining a therapeutic model of nursing[END_REF][START_REF] Smith | Caring and the Science of Unitary Human Beings[END_REF][START_REF] Boykin | The role of nursing leadership in creating caring environments in health care delivery systems[END_REF][START_REF] Brilowski | An evolutionary concept analysis of caring[END_REF], Finfgeld-Connett 2008). Although a plethora of theoretical approaches in the nursing literature attempt to define and analyze caring, this concept remains a largely unknown and covert component of professional nursing practice [START_REF] Leininger | Transcultural nursing: its progress and its future[END_REF][START_REF] Watson | Nursing: Human Science and Human Care[END_REF][START_REF] Bottoroff | Nursing: a practical science of caring[END_REF]. Empirical evidence has revealed R e v i e w C o p y incongruence between patients' and nurses' perceptions on the importance of nursing behaviours that convey caring; however, in the changing world of health care, it is important that nurses are able to define the parameters of their role and to ensure that such definitions are one with the views of the recipients of care.

There is still scarcity of literature and systematic evidence about how caring interventions can enhance patients' outcomes and help them to deal with the stress of illness more effectively [START_REF] Mayer | Oncology nurses' versus cancer patients' perceptions of nurse caring behaviors: a replication study[END_REF][START_REF] Cohen | Linking Nursing Care Interventions with Client Outcomes: A Community-Based Application of an Outcomes Model[END_REF]. However, there is a growing body of literature [START_REF] Johansson | Preoperative education for orthopedic patients: systematic review[END_REF][START_REF] Muller-Staub | Nursing diagnoses, interventions and outcomes -application and impact on nursing practice: systematic review[END_REF][START_REF] Suhonen | Hospitals' organizational variables and patients' perceptions of individualized nursing care in Finland[END_REF][START_REF] Suhonen | A review of outcomes of individualised nursing interventions on adult patients[END_REF] that explores how various nursing interventions, such as nursing assessment and patient education, can be beneficial to the patient. Previous reviews have described the concepts related to caring interventions and their efficacy on select patient outcomes, such as patient satisfaction and well-being [START_REF] Wolf | Relationship between nurse caring and patient satisfaction in patients undergoing invasive cardiac procedures[END_REF], Larrabee et al. 2004[START_REF] Green | Toward a predictive model of patient satisfaction with nurse practitioner care[END_REF][START_REF] Wu | Caring Behaviors Inventory: a reduction of the 42-item instrument[END_REF][START_REF] Raffii | Nurse caring in Iran and its relationship with patient satisfaction[END_REF].

The Review Aim

To test the hypothesis that nurses and patients perceive the concept of caring in nursing differently, by identifying the best available quantitative literature investigating nurse caring perceptions from the perspective of clients and nurses in a variety of settings. More specifically, this review aims to:

Examine congruence between patients' and nurses' perceptions of caring behaviours.

Identify areas of agreement and disagreement between these perceptions.

Design

For this quantitative comparative review, a narrative summary approach was adopted for reporting the findings. The methodology for the searching process followed the guidance of the Centre for Reviews and Dissemination for undertaking reviews (CRD 2009). The prespecified protocol included description of the research question, the review method, and the plan of how the data would be extracted and compiled. It was anticipated that such an approach would minimize the likelihood that the results or the expectations of the reviewing team would influence study inclusion or synthesis [START_REF] Garg | Systematic review and meta-analysis: when one study is just not enough[END_REF]. 2009, in order to locate the appropriate articles. Searching did not have any publishing time limit. Relevant studies were identified by using certain key words in different combinations. The same search strategy was adopted for each database. Search terms used were: care, caring, nurse, nursing, behavio(u)rs, patient, perception, quantitative, and comparative.

Search methods

R

Type of studies

Quantitative research studies, comparing both patients' and nurses' perceptions, published in scientific journals, which focussed on nurse-patient interaction and patient outcome in a variety of health care settings were included in the review. The authors do acknowledge the existence of a plethora of studies using the qualitative approach [START_REF] Sherwood | Meta-synthesis of qualitative analyses of caring: defining a therapeutic model of nursing[END_REF][START_REF] Patistea | Nurses' perceptions of caring as documented in theory and research[END_REF], Fingfeld-Connett 2008) but they were out of the scope of this review.

Inclusion criteria

These criteria included: Adult participants (18+ years of age), hospital or institutional settings, nurses, students and/or patients populations, quantitative research design, English language, and issues relevant to the study.

Search outcome

Phase one

Literature search was carried out by two members of the research team (E.P., G.E.) to ensure that all relevant articles would be located. Searching produced a total of 262 articles.

All went through a title screening by the same two members of the research team. Titles that both researchers agreed were irrelevant to the aim of this article were excluded. All other articles (98) that seemed relevant to the topic or for those that no consensus between the two researchers was reached were forwarded to the next phase. Duplicates were also considered.

Phase two

An evaluation of all abstracts of those articles that were selected at the previous phase followed. All abstracts were read and checked if they met the inclusion criteria. As previously, all studies that were agreed by the same two members of the team that met the criteria set were forwarded to the next phase. If no consensus was reached for a specific article, then this was also forwarded into the next phase. All other studies (47) were excluded. In the third and final phase of the search process, a total of 51 articles were read and compared to the inclusion criteria that were set. Additional hand searching of the list of references of all 51 articles did not produce any additional information. A final number of 34 articles were decided to be quality appraised.

Phase three

R

Quality appraisal

All 34 articles were carefully read and evaluated using a checklist designed by the authors adapted from existing literature [START_REF] Bowling | Research Methods in Health[END_REF], CRD 2009), and based on the requirements of the current study. Criteria used for considering a study methodologically sound are listed in Table 1. Every member of the research team had to respond with YES or NO, based on his/her judgement whether the appraised articles fulfilled each criterion (see supporting Table 1). For the studies that did not fulfil the inclusion criteria and were excluded from the review, please see supporting Table 2.

Following the quality appraisal process, a total of 11 articles were excluded (due to insufficient quality status) and a final number of 23 articles were used in this systematic literature review. To ensure that not a single article was mistakenly excluded, all full text excluded articles were read for a second time and reevaluated. None of these were found to be appropriate for the purposes of this study. Regarding the characteristics of the reviewed studies, see supporting information file 1. The process of identifying and including references for the systematic review is presented in figure 1.

Insert figure 1

Data abstraction and synthesis

Findings related to caring behaviours were extracted from the quality appraised research publications (Table 2). Extracted data included author(s) and date of publication, research hypothesis and aim(s) of the study, research instrument, used population and sample size, and findings (total scores of responses in order to compare perception). Data were extracted independently by two members of the research team (G.E., E.P.). Any disagreements were discussed and either resolved or the third member of the team (A.C.) offered an opinion and then decision was made. Synthesis of the extracted data was then performed following a narrative approach. Most studies were conducted in oncology units and others in rehabilitation centres [START_REF] Keane | Caring: Nurse-patient perceptions[END_REF] or long-term care settings [START_REF] Smith | Nurses' and patients' perceptions of most important caring behaviors in a long term care setting[END_REF], acute care units [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF]), psychiatric units (vonEssen & Sjöden 1993), medical-surgical units (Hegedus 1999), medical, surgical, and psychiatric units (vonEssen et al. 1994).

Insert table 2

Results

Settings

Instruments

It is interesting to note that in the early stages of this kind of research in the area of caring, data had been collected exclusively through the Q-methodology and the Care-Q instrument developed by [START_REF] Larson | Important nurse caring behaviors perceived by patients with cancer[END_REF], and this approach continued to influence research for many years. In this review, 18 out of the 23 studies used the Q-methodology or a modification and the rest used other instruments like the Caring Behaviors Inventory (CBI) [START_REF] Moyle | Validation of nurse caring behaviours in residential aged care[END_REF], the Caring Behavior Assessment (CBA) (O'Connell & Landers 2008) and the Caring Dimensions Inventory (CDI) (McCance et al. 2008). One study used the NCQ and PCQ (Nurse and Patient Caring Questionnaire) that consists of two parts which differ on the instructions, the first part asking nurses to state their preference regarding the importance of caring items, and the second to report on caring work actually performed [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF]. However, studies exploring the views of nurses separately of those of patients used a variety of caring instruments although there is a slight preference for using the Care-Q. There is also a tendency to use caring instruments in combination with other measures like patient anxiety and depression [START_REF] Larsson | Cancer patient and staff ratings of the importance of caring behaviors and their relations to patient anxiety and depression[END_REF][START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF], patient health and quality of life [START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF], or pain scales [START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF]).

The majority of the studies examined the most and least important nurse caring behaviours, whether patients and nurses differ in their ranking of those behaviours and if a significant relationship between nurses' and patients' perceptions of nurse caring behaviours exists.

One study explored whether nurses' and patients' gender has any effect on the importance of caring as well as on the presence of caring [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF], whereas other investigators examined the association within patient-nurses dyads regarding their perceptions of caring behaviours, anxiety, and depression [START_REF] Vonessen | Perceptions of caring behaviors and patient anxiety and depression in cancer patient-staff dyads[END_REF], Widmark-Peterson et al. 2000).

Differences in the dimensions of caring

Significant differences were found between patients and nurses in their perception of caring and caring behaviours in many of the reviewed studies. The Q methodology revealed marked differences between nurses and patients in the ranking of how important different nurse caring behaviours are considered to be. Patients appear to value the instrumental, technical caring skills more than nurses do, and perceive behaviours that demonstrate The mean values for Care-Q demonstrated that nurses assign a higher importance to 'comfort, anticipates, and trusting relationships' as significantly more important than the patients [START_REF] Larson | Comparison of cancer patients' and professional nurses' perceptions of important nurse caring behaviors[END_REF][START_REF] Mayer | Oncology nurses' versus cancer patients' perceptions of nurse caring behaviors: a replication study[END_REF][START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF][START_REF] Vonessen | Perceived importance of caring behaviors to Swedish psychiatric inpatients and staff, with comparisons to somatically-ill samples[END_REF][START_REF] Larsson | Cancer patient and staff ratings of the importance of caring behaviors and their relations to patient anxiety and depression[END_REF], Tuckett et al. 2009) and they consistently rank the 'Comfort' dimension as their first priority. Two studies have examined the perceptions as well as the occurrence of caring behaviours [START_REF] Vonessen | Perceived occurrence and importance of caring behaviours among patients and staff in psychiatric, medical and surgical care[END_REF][START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF]) assuming that perceptions may not be accurate predictors of what nurses actually do, but no significant association was demonstrated for nurses. Low correlations between patients mean values on subscale occurrence and importance was found in 'Explains and Facilitates', rated to be of high importance but of low frequency, whereas 'Comforts' was considered by surgical patients as the least important but relatively frequent [START_REF] Vonessen | Perceived occurrence and importance of caring behaviours among patients and staff in psychiatric, medical and surgical care[END_REF].

In several Care-Q studies, patients have considered the subscale 'Monitors and Follows Through' to be of high importance [START_REF] Larson | Comparison of cancer patients' and professional nurses' perceptions of important nurse caring behaviors[END_REF][START_REF] Mayer | Oncology nurses' versus cancer patients' perceptions of nurse caring behaviors: a replication study[END_REF][START_REF] Keane | Caring: Nurse-patient perceptions[END_REF][START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF], 1994[START_REF] Larsson | Cancer patient and staff ratings of the importance of caring behaviors and their relations to patient anxiety and depression[END_REF][START_REF] Tucket | Validation of CARE-Q in residential aged-care: rating of importance of caring behaviors from an e-cohort sun-study[END_REF]). This ranking differs from dyadic studies in which this caring dimension is ranked lower by both nurse R e v i e w C o p y and patient group (vonEssen & Sjoden 1991[START_REF] Vonessen | Perceptions of caring behaviors and patient anxiety and depression in cancer patient-staff dyads[END_REF][START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF]. The 'Explains and Facilitates' category is ranked high by patients and low by nurses in some studies [START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF], 1993[START_REF] Clifford | Caring: fitting the concept to nursing practice[END_REF], Widmark-Peterson et al. 2000), meaning that patients rate the information aspect as more important than nurses do and nurses stress the emotional aspect more than the patients. These authors have challenged the nurse communicative behaviours that may result in superficial assessment skills and planning care on own assumptions. Similar disparities are observed in the rating of items of the Care-Q and different perceptions were found in many of the items although both nurses and patients were quite diversified in selecting the most important items in terms of making them feel cared for.

Patients chose more instrumental behaviours like 'knows how to give shots, IVs, and manage equipment' while nurses choose more expressive behaviours like the item 'listens to the patient' [START_REF] Larson | Comparison of cancer patients' and professional nurses' perceptions of important nurse caring behaviors[END_REF][START_REF] Mayer | Oncology nurses' versus cancer patients' perceptions of nurse caring behaviors: a replication study[END_REF][START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF], Scharf & Caley 1991[START_REF] Gooding | Important nurse caring behaviors: perceptions of oncology patients and nurses[END_REF], O'Connell & Landers 2008). Analogous results were found with the use of other instruments with the exception of one that found no significant differences between the patient and nurses groups [START_REF] Moyle | Validation of nurse caring behaviours in residential aged care[END_REF]. However, the validity of this later study is compromised by the very small sample (16 nurses and 31 residents) and there are concerns regarding patient samples drawn from old peoples' homes, mainly because of the difficulties in engaging in a Q sort procedure, and the high dependency of patients on nursing care and the feeling of obligation towards the nursing staff.

Patients with different kind of health problems

Cancer patients are the most frequently studied patient group in the area of caring and the results to whether the cancer trajectory influences the perceptions of both patients and nurses are contradictory. It is assumed that the cancer patients and nurses may establish a long-term care relationship so they may develop more consistent perceptions regarding the importance of caring behaviours. It is also possible that because of the symptoms like pain and suffering, patients might need more frequent contact, monitoring, and follow-up from nurses than patients with other diseases. This assumption [START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF] would generate expectations that patients with cancer and oncology nurses do not differ in their perceptions of caring behaviours. However, this assertion was partially supported, especially in studies that, opposed to others, paired staff with patients on the same unit.

Similarly, studies conducted in specialties of hematological malignancies [START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF] reported a higher ranking of the 'Monitors and Follows Through' R e v i e w C o p y category among nurses and congruence with the patients' ratings, in contrast to studies in which nurses gave a lower ranking to this specific subscale [START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF][START_REF] Vonessen | Perceptions of caring behaviors and patient anxiety and depression in cancer patient-staff dyads[END_REF]) leading to the hypothesis that in areas of very advanced treatments, nurses also perceive their technical role as more important.

In contrast to the finding that patients rank physical care higher than nurses do (vonEssen & Sjoden 1993), in the study of psychiatric inpatients and staff, it was found that patients with mental health problems consider the cognitive aspect of caring as the most important aspect of care. A comparison of patients with somatic problems in the same study showed that somatic and psychiatric patients differ in their perception of caring behaviours, since the task-oriented aspect of caring was rated as more important from patients hospitalized in medical and surgical settings.

Another area of differentiation is the critical care settings, where a low ranking was given to caring subscales like a trusting relationship, expression of positive feelings, and existential feelings (O'Connell & Landers 2008), suggesting that interpersonal nursepatient relationships may not be considered very important in this area of practice.

Interesting differences and similarities are also reported by Eksrom (1999) who found that patients' expectations regarding caring behaviours are higher when the nurse is a female, and lower when the nurse is a male, leading to the assumption that gender stereotypes may affect perception of behaviour in others.

Caring behaviours and patient outcomes

Nursing outcomes describe changes in a patient's state of health as a result of nursing interventions like changes in functional status, coping, or self-care [START_REF] Muller-Staub | Nursing diagnoses, interventions and outcomes -application and impact on nursing practice: systematic review[END_REF]. There is scarcity of research that relates nursing behaviours to patient outcomes, although studies are proliferating concerning the benefits of certain interventions like nursing assessment and diagnosis [START_REF] Muller-Staub | Nursing diagnoses, interventions and outcomes -application and impact on nursing practice: systematic review[END_REF], patient education [START_REF] Johansson | Preoperative education for orthopedic patients: systematic review[END_REF], preventative interventions or caring approaches like individualized care [START_REF] Suhonen | Hospitals' organizational variables and patients' perceptions of individualized nursing care in Finland[END_REF]). More studies exploring caring behaviours are focused on outcomes in terms of patient satisfaction [START_REF] Wolf | Relationship between nurse caring and patient satisfaction[END_REF][START_REF] Wolf | Relationship between nurse caring and patient satisfaction in patients undergoing invasive cardiac procedures[END_REF], Larrabee et al. 2004[START_REF] Green | Toward a predictive model of patient satisfaction with nurse practitioner care[END_REF][START_REF] Wu | Caring Behaviors Inventory: a reduction of the 42-item instrument[END_REF][START_REF] Raffii | Nurse caring in Iran and its relationship with patient satisfaction[END_REF]) but they were not within the scope of this research, in spite of the interesting correlations found between caring behaviours and general satisfaction of patients from caring.

Two studies explored the discrepancies between nurse and patient perceptions of the importance of caring behaviours and patient level of anxiety and depression, assuming that R e v i e w C o p y they have potentially important consequences on the quality of caring [START_REF] Vonessen | Perceptions of caring behaviors and patient anxiety and depression in cancer patient-staff dyads[END_REF], Widmark-Peterson et al. 2000). Neither significant mean differences, nor significant correlations between patients and nurses were found on the hospital anxiety and depression levels and the members of the matched nurse-patient dyads did not agree strongly on the importance of caring behaviours [START_REF] Vonessen | Perceptions of caring behaviors and patient anxiety and depression in cancer patient-staff dyads[END_REF], Widmark-Peterson et al. 2000). The authors support that these results which are contrary to previous findings, suggest that nurses are not sensitive enough to their patients' reported levels of anxiety and depression, do not provide sufficient opportunities for communication, or fail to recognize the patients' emotional state. Patients' and nurses' perceptions about the patients' health quality of life, greatest health-related concern and satisfaction with caring were explored in addition to the patients and nurses perceptions of caring behaviours by Widmark-Peterson et al. (2000). No correlations between nurses' and patients' perceptions of these variables were found indicating that nurses are not aware of what aspects of caring patients consider important in order to feel well-cared for, nor of the individual patients' health, related concerns, or quality of life. In the case of pain, it seems that patients with higher levels of pain receive more supportive nurse caring and patients with the most intense pain rated higher on perceived caring behaviours than others [START_REF] Wu | Caring Behaviors Inventory: a reduction of the 42-item instrument[END_REF]. A statistically significant correlation of pain and caring behaviours, either positive or negative, was also reported with the use of different instruments [START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF].

Discussion

Limitations

Research comparing both nurses' and patients' perceptions of caring using the quantitative approach only, was included in this study. Although some evidence suggesting that there is no congruence between nurses and patients perceptions on the concept of caring in nursing has been established, a more synthesised approach should be attempted, including both quantitative and qualitative research studies in order to examine the issue in more depth.

The number of the well-designed studies included in this review is quite limited but a less rigid set of design criteria would have undermined the validity of the evidence produced [START_REF] Johansson | Preoperative education for orthopedic patients: systematic review[END_REF]. This means that the conclusions drawn are necessarily tentative. Specific inclusion criteria were used in order to demonstrate the current evidence in the field. Had studies dealing with the broader perception of caring or studies with samples either from nurses or patients been included, it would have been feasible to offer more R e v i e w C o p y generalizable information; on the other hand, we would have detracted from the main purpose of this review. Some studies included in this review used small samples. The authors, although they acknowledge this as a study limitation, have decided to include them in this systematic review in the absence of studies conducted with larger samples. Studies with limited samples were only included if they fulfilled all the other pre-specified criteria of quality status.

Methodological considerations

Questions are raised as to how caring is defined and measured. Most studies used operational definitions and behavioural designations to the scientific study of caring, because as it was stated by [START_REF] Wolf | Dimensions of nurse caring[END_REF], it is through the practice of caring behaviours, including acts, conduct and mannerisms that nurses convey caring and the feeling of being cared for. On the other hand, it has been criticised that the persistence to caring as an elusive and non-measurable concept, has inhibited nursing's scientific development [START_REF] Paley | Caring as a slave morality: Nietzchean themes in nursing ethics[END_REF]. Many elusive concepts in other disciplines have been described and measured. Good examples are empathy [START_REF] Yu | Evaluation of empathy measurement tools in nursing: systematic review[END_REF] or depression for which psychologists have developed strong and reliable instruments to quantify and measure which are used not only as research tools but also as partially diagnostic instruments.

Although each research design offers its own contribution to the increase of nurses' understanding on caring, methods that will allow us to describe and quantify nursing's unique contribution to health care and link caring with patients' outcomes, as well as procedures that stand the scientific scrutiny need to be developed further.

Result of the review

There is some evidence of the assertion that there is no congruency of perceptions between patients and nurses as regards which behaviours are considered caring; although, in some studies, aspects of congruence have also been reported. The caring model that was complex and technical caring environment in which caring may be seen as losing the affective aspects as well by the fact that they need to deal with the shortages of both nurses and time as consequences of a tight economy. However, none of the studies reviewed described other environmental parameters or factors surrounding caring and nurse-patient interactions that could determine the caring element of nursing. Only a few studies attempted to explore the correlation between caring behaviours in producing therapeutic client-caring outcomes which would explain the nurses' unique contribution to caring.

A considerable amount of research on caring in this review has emerged from investigations on oncology units, and is limited to the secondary level of prevention where both patient and nurse caring perceptions might be determined due to organizational and other influences. The increasing complexity of caring and patient needs, reduced hospital lengths of stay, and reduced ratio of nurses to patients may influence nurses to take a less holistic approach to nurse caring (Larrabee et al. 2004). In addition, it may lead patients to seek a more particularistic approach. On the other hand, the patients' emphasis on the technical aspect of caring, such as proficiency in giving medications, managing intravenous fluids, 'Monitors and Follows Through' are the needs of highly vulnerable, dependent persons who have had to place their hopes in the hands of somebody else; in this case, in the hands of a nurse [START_REF] Rosenthal | Coronary patients' and nurses' perceptions of important nurse caring behaviors[END_REF].

Applicability, theoretical and practical implications of the findings

The results of this review make a contribution to the debate concerning the definition of caring from the perspective of specific behaviours that convey caring and fulfil patients' expectations of nurse caring. There is a need to promote a patient-focused philosophy [START_REF] Suhonen | A review of outcomes of individualised nursing interventions on adult patients[END_REF]) and develop a common understanding of caring to improve caregiverpatient interaction, to plan, implement, and evaluate caring that is not based on assumptions. In order to plan care that responds to every patient's individualized situation and needs, nurses have to elicit and use individual patients' preferences more systematically in care planning.

Further research is needed to generate more knowledge on the relationship between caring behaviours, patient outcomes and health or nursing costs, as well as significant elements of the caring environment. There is a growing body of literature suggesting that congruency of perceptions and goals is important for the patients profiting from caring, and nurse-patient agreement may be the key factor in patient satisfaction and consequently patient's recovery, comfort, health behaviours, and compliance. An implication to hospital leaders is to R e v i e w C o p y monitor patient-perceived nurse caring because of its demonstrated relationship with patient satisfaction with nursing care, the key predictor of patient satisfaction with hospital care (Larrabee et al. 2004). Health care organizations need to take into account the patients' perceptions on caring and introduce restructuring of hospital systems aiming at reducing labour costs through work redesign. Information given in this paper may be also useful in planning educational programmes for students and practicing nurses, and contributes empirical evidence towards the body of knowledge related to caring behaviours.

Conclusion

This review demonstrates that nurses do not always accurately assess patient perceptions on the importance of various dimensions of caring which means that they may plan and implement caring for the patient based on their own assumptions. Such information is valuable because the professional responsibility of nurses is centred on providing high quality nursing interventions leading to positive outcomes [START_REF] Suhonen | A review of outcomes of individualised nursing interventions on adult patients[END_REF]. This

R e v i e w C o p y

Convenience A t-test for all the items showed significant differences for 14 out of the 50 items in the Care Q between patients and nurses. On the questionnaire form, nurses gave significantly higher values than patients to 30 items. [START_REF] Rosenthal | Coronary patients' and nurses' perceptions of important nurse caring behaviors[END_REF] To

Convenience

Significant nurse-patient differences were found in 13 of the 50 items (3 items p<0.001, 3 items p<0.005, 5 items p<0.01 and 2 items p<0.05). Patients rated the cognitive aspect of care as more important, while nurses stress the emotional component. Both groups regard a trusting relationship and the emotional aspect of care as highly important in order to make patients feel cared for. [START_REF] Gooding | Important nurse caring behaviors: perceptions of oncology patients and nurses[END_REF] 1) How do oncology nurses and patients rank caring behaviors in order of importance 2) What is the relationship between these rankings 3) Are there differences between the subscales between nurses and patients Care -Q 42 oncology patients 46 oncology nurses Convenience Two-tailed t-tests revealed that there were significant differences between the patients and nurses with respect to the ranking of the subscales of clinical caring (p<0.001), Disposition of Nurse Caring (p<0.001) and Continuity of Nurse Caring (p<0.001).

Von (patients>staff, p<0.05) and «accessible» (staff>patient, p<0.01). A significant negative correlation was found for "comfort" (p<0.05) but no between-group difference Patient and nurses mean values differed significantly in 10 of the 50 items (t-test, six items p<0.01, four items p<0.05). Significant correlations were found for three items (p<0.05).

Generally, mean values demonstrated disparities for 2 of the 6 subscales and 10 of the 50 items and negative correlations indicating opposing views within the nurse-patient dyads.

Von Essen & Sjoden 1995

To investigate the perceptions of in-patient and staff (nurse) on the occurrence and importance of caring behaviours and the occurrence that these behaviours occur Care -Q Questionnaire of the Care -Q In-patients receiving psychiatric (61), medical (47) and surgical care (40) Nursing staff in psychiatric (63), medical (43) and surgical care (27) Nurse aides, nurse assistants and nurse students included in the above nursing samples Convenience Two-way ANOVAS yielded differences (occurrence) in the "comfort" "accessible" "anticipates" and "trusting relationship" subscales, where nurses considered caring behaviors belonging to these subscales to have been more frequent than patients did. Significant patient group interactions observed for "comfort" (p<0.05) with higher values from staff than from patients in psychiatric and medical care but not in surgical care. The care-Q (importance subscale) showed significant nursepatient differences Widmark-Peterson et al. 1996 To investigate a) whether cancer patients and staff have different cognitive representations of the concept caring and clinical care b) whether 2 different wordings of the response categories used by patients and staff to rank the perceived importance of 50

Care -Q 72 cancer patients 63 staff Convenience

On the subscales values significant differences were found in accessibility (patients scored lower than staff, p<0.05), comfort (nurses gave higher value than patients, p<0.01) and trusting relationship (nurses gave higher means than patients, p<0.01).

In the rank ordering of the subscales, differences were found in both forced and free response format questions, where nurses perceive expressive affective behaviors as most important, whereas the most important behaviors as perceived by patients Statistically significant differences between patients' and nurses' perceptions were found in 14 of the 50 statements (p<0.05) using a t test for differences in means for independent samples. The mean value for the item "listens to the patient "

R e v i e w C o p y

was for nurses 5.7 (rank 1) and for patients 4.9 (rank 3) (p=0.001) [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF] When nurses respond, is there a main effect due to gender of the nurse on the importance of caring and on the presence of caring?

The second question is for the patients' response and the effect due to the gender of the nurse.

Convenience

There were no significant correlations between patients (CARE-P) and nurses (CARE-S) perceptions of the importance of the Care-Q subscales or significant associations between patient (CARE-P) perceptions and nurse views of patients' perceptions (CARE-SP). CARE-S and CARE-SP were significantly positively associated with 4 subscales: Accessible (p<0.01), Comforts (p<0.01), Anticipates (p<0.01) and Trusting Relationship p<0.01). Patient levels of anxiety, patients health and quality of life did not correlate with their ratings of the CARE-Q subscales [START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF] To explore differences in perceived importance of nursing The results demonstrated that there are more similarities than differences between the perceptions of nurses and relatives with 6 of the most important items common to both groups and in the ranking order of the subscales. Both groups placed a higher value on caring behaviors which demonstrate technical competence, altruistic and emotional aspects of caring Tuckett et al. (2009) To demonstrate that the free response form of the 50-item care Q is acceptable to nurses and residents in nursing homes.

Care-Q questionnaire form 3 aged care facilities 37 long-term residents 90 nurses

The subscale comfort was rated significantly lower by the patients (M=6.3) as related to nurses (M=6.8) p=0.003. Patients also rated lower in the subscales Anticipates (M=6.2) as related to nurses (M=6.4) p=0.009 and Trusting relationship

R e v i e w C o p y

Characteristics of the reviewed studies

All studies had a clear statement of research question(s) and almost all researchers used an operational definition of caring, that is of caring behaviors. The scientific background and explanation of the rationale was reported in all the documents except two (Keane et al. 1987, Smith and[START_REF] Smith | Nurses' and patients' perceptions of most important caring behaviors in a long term care setting[END_REF]. Most of the instruments used were based on caring theory and their psychometric properties have been published [START_REF] Watson | Assessing and Measuring Caring in Nursing and Health Science[END_REF]. Very few of the reviewed studies have presented validation analyses for their sample and this included mainly a Cronbach's alpha coefficient [START_REF] Huggins | Emergency department patients' perception of nurse caring behaviors[END_REF][START_REF] Wolf | Relationship between nurse caring and patient satisfaction[END_REF][START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF][START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF], O' Connel & Landers 2008[START_REF] Tucket | Validation of CARE-Q in residential aged-care: rating of importance of caring behaviors from an e-cohort sun-study[END_REF]. [START_REF] Gooding | Important nurse caring behaviors: perceptions of oncology patients and nurses[END_REF] in an effort to explore further the groupings of the nurse caring behaviors identified five new subscales of the Care-Q, validated them through a review panel and removed 21 items.

A convenience sampling was used in all the papers reviewed and no study has reported a power analysis for the sample size. The number of participants ranged from 14 patients and 15 nurses [START_REF] Smith | Nurses' and patients' perceptions of most important caring behaviors in a long term care setting[END_REF] to 145 patient -nurse dyads [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF]) and the type of sampling was convenient in all studies. Only two studies used matched pairing techniques for their samples [START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF][START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF]. Two studies included relatives (McCance et al. 2008, O' Connel & Landers 2008), one study reported a comparison of patients, nurses and doctors [START_REF] Scharf | Patients', nurses', and physicians' perceptions of nurses' caring behaviors[END_REF] and another used psychiatric and somatically ill patient samples [START_REF] Vonessen | Perceived importance of caring behaviors to Swedish psychiatric inpatients and staff, with comparisons to somatically-ill samples[END_REF][START_REF] Clifford | Caring: fitting the concept to nursing practice[END_REF]. Inclusion criteria were only used for the settings, such as medical, surgical or other wards. Eligibility criteria for the participant nurses were not described and most studies used only registered nurses although some researchers included students, nurse assistants and nurse aides working on the ward [START_REF] Vonessen | The importance of nurse caring behaviors as perceived by Swedish hospital patients and nursing staff[END_REF], 1993[START_REF] Clifford | Caring: fitting the concept to nursing practice[END_REF][START_REF] Widmark-Petersson | Cancer patient and staff perceptions of caring and clinical care in Free versus Forced choice response formats[END_REF], 2000[START_REF] Moyle | Validation of nurse caring behaviours in residential aged care[END_REF][START_REF] Tucket | Validation of CARE-Q in residential aged-care: rating of importance of caring behaviors from an e-cohort sun-study[END_REF]). More detailed eligibility criteria for the patient sample were used in specialized areas like cardio and oncology and included a histological diagnosis of cancer [START_REF] Larson | Comparison of cancer patients' and professional nurses' perceptions of important nurse caring behaviors[END_REF], patients receiving chemotherapy [START_REF] Mayer | Oncology nurses' versus cancer patients' perceptions of nurse caring behaviors: a replication study[END_REF]) alertness, freedom of chest pain and physician's consent [START_REF] Scharf | Patients', nurses', and physicians' perceptions of nurses' caring behaviors[END_REF], experience of pain and pain medication in the last 24 hours [START_REF] Chang | Cancer patient and staff ratings of caring behaviors: relationship to level of pain intensity[END_REF]. Some studies excluded patients being senile, mentally or emotionally disturbed or with limited vision and hearing [START_REF] Vonessen | Perceived importance of caring behaviors to Swedish psychiatric inpatients and staff, with comparisons to somatically-ill samples[END_REF][START_REF] Clifford | Caring: fitting the concept to nursing practice[END_REF][START_REF] Larsson | Cancer patient and staff ratings of the importance of caring behaviors and their relations to patient anxiety and depression[END_REF], 2000) or fatigue [START_REF] Widmark-Petersson | Perceptions of caring among patients with cancer and their staff. Differences and disagreements[END_REF] whereas others used exclusively elderly patients [START_REF] Moyle | Validation of nurse caring behaviours in residential aged care[END_REF], Tuckett et al. 2009). ANOVAs were used in a study of caring and gender [START_REF] Ekstrom | Gender and perceived nurse caring in nurse-patient dyads[END_REF].

The interpretations of the findings were clearly presented in all the documents reviewed but generalisability was not supported by any of the studies reviewed because of the convenient or small samples, lack of homogeneity and arbitration in the selection criteria.

The twenty three research documents involved 1229 patients and 1390 nurses. Most of the studies were conducted in the USA (n=11), in Canada (n=1), Australia (n=2)

Taiwan (n=2) and the majority of the European were conducted in Sweden (n=7).

However in the Swedish language, two different concepts that is of "caring" and "clinical care" are described and patients may have different cognitive representations of caring, therefore any generalisability to other European cultures should be made with caution (Widmark-Peterson et al. 1996[START_REF] Gardner | Don't call me sweetie!'Patients differ from nurses in their perceptions of caring[END_REF]).

  generate more knowledge on the relationship between caring behaviours, patient outcomes and health or nursing costs.

  in the literature as the central focus of nursing. It has been considered

  , and EMBASE were thoroughly searched between March and May

  to perform nursing activities ('know how') as more important. On the other hand, nurses perceive their psychological skills and expressive or affective caring behaviour as more important than patients do, leading to the conclusion that nurses may misperceive the necessity of the emotional aspect of caring in comparison with patient judgments. These results, repeatedly reported in the research literature, indicate that nursing staff may not accurately assess patient perceptions of caring and that patient care is not congruent to the patients' preferences, expectations, or individual needs.The Care-Q that is the most frequently used instrument ranks caring behaviours in six categories as follows: Accessible, Explains and Facilitates, Comforts, Anticipates, Trusting Relationship, and Monitors and Follows Through. Other instruments reviewed describe similar subscales, for example, the CBI (Assurance of Human Presence, Professional Knowledge and Skill, Respectful Deference to Others, and Positive Connectness/Attentiveness to Other's Experience) and the CBA (Humanism, Helping/Trusting, Expression of Feelings, Teaching/Learning, Supportive and Protective Environment, Human Needs Assistance and Existential, Phenomenological, Spiritual forces) or less similar subscales, for example, the CDI (Psychosocial, Technical, Professional, Inappropriate, and Unnecessary Nursing Activities). The majority of these studies reported significantly different nurse-patient perceptions of caring, something that is demonstrated both on analyses of the subscales as well as on the individual items.

  extensively used in this review describes two categories of caring interventions that are both expressive and instrumental in nature. Expressive activities include establishing trusting relationships and offering support, whereas instrumental activities include physical action-oriented helping behaviours and cognitively oriented helping behaviours. Patients appear to value the instrumental, technical skills more than nurses do, which means that they may not be open or receptive to the expressive caring behaviours until basic physical needs are met through instrumental activities. On the other hand, the psychological orientation of nurses (Widmark-Petersson et al. 1998) may be explained by the very

Figure 1 :

 1 Figure 1: Flow diagram of the process of identifying and including references for the systematic review

  All researchers used descriptive statistics such as mean scores, standard deviations, frequencies, percentages, maximum and minimum scores. Statistical comparisons of patient and nurses group means were performed by using t-test, whereas Pearson, Wilcoxon and Spearman rank correlation coefficients were employed for the analysis of associations and differences between patients and staff. A 3-way ANOVAs and the Mann-Whitney U-test were used in one study to compare patient and staff subscale scores, within and between scores[START_REF] Larsson | Cancer patient and staff ratings of the importance of caring behaviors and their relations to patient anxiety and depression[END_REF] and 2-way

Table 2 :

 2 Summary of the studies reviewed

	Convenience	(M=6.2) as related to nurses (M=6.5) p=0.049.
	To validate the factors

Journal of Advanced Nursing

R e v i e w C o p y

Positive significant correlation between patients and nurses evaluation on important behaviours (p<0.01). Significant agreement between nurses and patients choices for the three of the six major categories of caring behaviors: explains and facilitates (r=.81, p= .0499), Trusting relationships (p=0 .0174) and Monitors and Follows Through (p=0 .0427). The correlations for the other three categories were not significant.

The differences between patient and nurse ranked behaviors showed only "is cheerful" as statistically significant (p≤0.5) with patients ranking this behavior as more important. Keane et al.1987 To identify perceptions of patients and nurses of the most and least important nurse caring behaviors. Significant differences among patient and nurses on five out of the six scales of the Care-Q (Patients assigned higher values to the Explains and Facilitates, p=0.004 and Monitors and Follows through p=0.0046 and Nurses scored higher on the Comforts subscale, p=0.0001). In the questionnaire form, Nurses gave higher values to the Accessible, p=0.0013, Comforts, p=0.0001, Anticipates, p=0.0001, and Monitors and Follows through p=0.0002

Care

Author(s)

Reason for exclusion


Olav Geil

Casper Thomsen
email: caspert@math.aau.dk

List decoding of a class of affine variety codes

Keywords: Affine variety code, list decoding, multiplicity, Schwartz-Zippel bound

Consider a polynomial F in m variables and a finite point ensemble S = S1 × • • • × Sm. When given the leading monomial of F with respect to a lexicographic ordering we derive improved information on the possible number of zeros of F of multiplicity at least r from S. We then use this information to design a list decoding algorithm for a large class of affine variety codes.

Introduction

In this paper we study affine variety codes over F q in the special case where the variety is S 1 × • • • × S m . Here, S i ⊆ F q , i = 1, . . . , m. More formally write S = S 1 × • • • × S m = {P 1 , . . . , P |S| } and consider the evaluation map ev S : F q [X 1 , . . . , X m ] → F |S| q , ev S (F ) = (F (P 1 ), . . . , F (P |S| )). Clearly, the generalized Reed-Muller codes is a special example of such a code. Other well-known examples are the hyperbolic codes [START_REF] Geil | On Hyperbolic Codes[END_REF], Reed-Solomon product codes [START_REF] Augot | List decoding of Reed-Solomon product codes[END_REF], (generalized) toric codes [START_REF] Hansen | Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes[END_REF][START_REF] Ruano | On the structure of generalized toric codes[END_REF][START_REF] Bras-Amors | Duality for some families of correction capability optimized evaluation codes[END_REF], and Joyner codes [START_REF] Joyner | Toric Codes over Finite Fields[END_REF]. In the present work we show that the dimension of E(M, S) equals |M| and we present a lower bound on the minimum distance which turns out to be sharp when |M| satisfies certain reasonable criteria. We then present a decoding algorithm for E(M, S). The algorithm is a straightforward generalization of the Guruswami-Sudan decoding algorithm for Reed-Solomon codes [START_REF] Guruswami | Improved decoding of Reed-Solomon and algebraicgeometry codes[END_REF], except that it involves a preparation step. A main ingredient in the preparation step is information about how many zeros of multiplicity at least r a polynomial F (X 1 , . . . , X m ) can have given information about its leading monomial with respect to a lexicographic ordering.

Let M ⊆ {X i1 1 • • • X im m | 0 ≤ i j < |S j |,
It is well-known that the generalized Reed-Muller codes can be viewed as subfield subcodes of Reed-Solomon codes. As demonstrated by Pellikaan and Wu [START_REF] Pellikaan | List Decoding of q-ary Reed-Muller Codes[END_REF][START_REF] Pellikaan | List Decoding of q-ary Reed-Muller Codes[END_REF] this observation leads to an efficient decoding algorithm of generalized Reed-Muller codes via the Guruswami-Sudan decoding algorithm for Reed-Solomon codes. When the variety does not equal S = F q × • • • × F q , or when M is not chosen as all the monomials of degree up to some number, such a neat result does not hold any more. In the present work we improve upon the methods from [START_REF] Dvir | Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers[END_REF] to derive more detailed information on how many zeros of prescribed multiplicity a polynomial can have. Rather than using only information on the degree of the polynomial we use information on the leading monomial with respect to a lexicographic ordering. We present both a recursive algorithm to find such bounds and also closed formula expressions for the case of polynomials in two variables. The interpolation polynomial Q(X 1 , . . . , X m , Z) in the list decoding algorithm of the present paper can be viewed as a polynomial in Z with coefficients from

F q [X 1 , . . . , X m ].
Having fixed the code to be used, in a preparation step we determine sets from which the coefficients must be taken. Here, we use our improved insight on how many zeros with prescribed multiplicity a polynomial can have given information about its leading monomial. The idea of a preparation step comes from [START_REF] Geil | Generalized Sudan's list decoding for order domain codes[END_REF] where an interpretation of the Guruswami-Sudan decoding algorithm without multiplicity was described for order domain codes. Our experiments show that the method of the present work is an improvement upon the situation where only the generalized Schwartz-Zippel bound is used for the design and the analysis of the decoding algorithm. Our algorithm works for codes of not too high dimensions. For small dimensions the algorithm often decodes more than d/2 errors.

Parameters of the codes

Throughout the paper we shall use the notation s i = |S i | for i = 1, . . . , m. Also we shall write n = |S| as clearly the length of E(M, S) equals |S|. First we show how to find the dimension of the code.

Proposition 1. The dimension of E(M, S) equals |M|.

Proof. We show that {ev S (X i1 1 , . . . , X im m ) | 0 ≤ i j < s j , j = 1, . . . , m} constitutes a basis for F n q as a vectorspace over F q . For this purpose it is sufficient to show that the restriction of ev to

{G(X 1 , . . . , X m ) | deg Xi (G) < s i , i = 1, . . . , m} (1)
is surjective. Given (a 1 , . . . , a n ) ∈ F n q let

F (X 1 , . . . , X m ) = n v=1 a v m i=1 a∈Fq\{P (v) i } X i -a P (v) i -a .
Here, we have used the notation

P v = (P (v) 1 , . . . , P (v)
n ), v = 1, . . . , n. It is clear that ev S (F ) = (a 1 , . . . , a n ) and therefore ev S : F q [X 1 , . . . , X m ] → F n q is surjective. Consider a monomial ordering. Let R(X 1 , . . . , X m ) be the remainder of F (X 1 , . . . , X m ) after division with {A 1 (X 1 , . . . , X m ), . . . , A m (X 1 , . . . , X m )}. Here, A i (X 1 , . . . , X m ) = a∈Si (X i -a), i = 1, . . . , m. Clearly, F (P i ) = R(P i ) = a i , i = 1, . . . , n. Hence, the restriction of ev S to (1) is indeed surjective.

We next show how to estimate the minimum distance of E(M, S). The Schwartz-Zippel bound [START_REF] Schwartz | Fast probabilistic algorithms for verification of polynomial identities[END_REF][START_REF] Zippel | Probabilistic algorithms for sparse polynomials[END_REF][START_REF] Demillo | A Probabilistic Remark on Algebraic Program Testing[END_REF] is as follows:

Theorem 1. Let X i1 1 • • • X im m be the leading monomial of F (X 1 , . . . , X m ) with respect to a lexicographic ordering. The number of elements in S = S 1 × • • • × S m that are zeros of F is at most i 1 s 2 • • • s m + s 1 i 2 s 3 • • • s m + • • • + s 1 • • • s m-1 i m .
The proof of this result is purely combinatorial. Using the inclusion-exclusion principle it can actually be strengthened to the following result which is a special case of the footprint bound from Gröbner basis theory: Theorem 2. Let notation be as in Theorem 1. The number of elements in S that are zeros of

F is at most equal to n -(s 1 -i 1 )(s 2 -i 2 ) • • • (s m -i m ). Proposition 2. The minimum distance of E(M, S) is at least min{(s 1 -i 1 )(s 2 -i 2 ) • • • (s m -i m )|X i1 1 • • • X im m ∈ M}.
The bound is sharp if for every M ∈ M all divisors of M also belong to M.

Proof. The first part follows from Theorem 2. To see the last part write for i = 1, . . . , m, S i = {b

(i) 1 , . . . , b (i) |Si| }. Clearly, F (X 1 , . . . X m ) = m v=1 iv j=1 X v -b (v) j has leading monomial X i1 1 • • • X im m
with respect to any monomial ordering and evaluates to zero in exactly n -(s

1 -i 1 )(s 2 -i 2 ) • • • (s m -i m ) points from S.
Finally, any monomial that occurs in the support of F is a factor of

X i1 1 • • • X im m .
3 Bounding the number of zeros of multiplicity r

In the following let X = (X 1 , . . . , X m ) and T = (T 1 , . . . , T m ).

Definition 1. Let F be any field. Given F (X) ∈ F[X] and k ∈ N m 0 then the k'th Hasse derivative of F , denoted by F (k) (X), is the coefficient of T k in F (X + T ). In other words F (X + T ) = k F (k) (X)T k .
The concept of multiplicity for univariate polynomials is generalized to multivariate polynomials in the following way: Definition 2. For F (X) ∈ F[X]\{0} and a ∈ F m we define the multiplicity of F at a denoted by mult(F, a) as follows: Let r be an integer such that for every

k = (k 1 , . . . , k m ) ∈ N m 0 with k 1 + • • • + k m < r, F (k) (a) = 0 holds, but for some k = (k 1 , . . . , k m ) ∈ N m 0 with k 1 + • • • + k m = r, F (k) (a) = 0 holds, then mult(F, a) = r. If F = 0 then we define mult(F, a) = ∞.
Elaborating on the results in [START_REF] Dvir | Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers[END_REF] we find:

Theorem 3. Let F (X) ∈ F[X] be a non-zero polynomial and let X i1 1 • • • X im m
be its leading monomial with respect to a lexicographic ordering. Then for any finite sets S 1 , . . . , S m ⊆ F

a∈S1ו••×Sm mult(F, a) ≤ i 1 s 2 • • • s m + s 1 i 2 s 3 • • • s m + • • • + s 1 • • • s m-1 i m .
Theorem 1 generalizes to Corollary 1 which we call the Schwartz-Zippel bound.

Corollary 1. Let F (X) ∈ F[X] be a non-zero polynomial and let X i1 1 • • • X im m be its leading monomial with respect to a lexicographic ordering. Assume S 1 , . . . , S m ⊆ F are finite sets. Then over S 1 × • • • × S m the number of zeros of multiplicity at least r is less than or equal to

i 1 s 2 • • • s m + s 1 i 2 s 3 • • • s m + • • • + s 1 • • • s m-1 i m /r. (2)
Just as we were able to improve upon Theorem 1 by using the inclusion-exclusion principle we will also be able to improve upon Corollary 1. However, now the situation is much more complex and therefore the inclusion-exclusion principle is no longer sufficient. What we need to strengthen Corollary 1 is the following rather technical function:

(s m -u 1 -• • • -u r )D(i 1 , . . . , i m-1 , r, s 1 , . . . , s m-1 ) + u 1 D(i 1 , . . . , i m-1 , r -1, s 1 , . . . , s m-1 ) + • • • + u r-1 D(i 1 , . . . , i m-1 , 1, s 1 , . . . , s m-1 ) + u r s 1 • • • s m-1
where

A(i m , r, s m ) = {(u 1 , . . . , u r ) ∈ N r 0 | u 1 + • • • + u r ≤ s m and u 1 + 2u 2 + • • • + ru r ≤ i m }. Theorem 4. For a polynomial F (X) ∈ F[X] let X i1 1 • • • X im m be its leading monomial with respect to the lexicographic ordering with X m ≺ • • • ≺ X 1 .
Then F has at most D(i 1 , . . . , i m , r, s 1 , . . . , s m ) zeros of multiplicity at least r in S 1 × • • • × S m . The corresponding recursive algorithm produces a number that is at most equal to the number found in Corollary 1 and at most equal to

s 1 • • • s m .
Proof. The proof involves a modification of the method described in [START_REF] Dvir | Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers[END_REF]. Remark 1. Given (i 1 , . . . , i m , r, s 1 , . . . , s m ) with ⌊i 1 /s 1 ⌋+• • •+⌊i m /s m ⌋ ≥ r then there exist polynomials with the leading monomial being X i1 1 • • • X im m such that all points in S 1 × • • • × S m are zeros of multiplicity at least r. Hence, we need only apply the algorithm to tuples (i 1 , . . . , i m ) such that

⌊i 1 /s 1 ⌋ + • • • + ⌊i m /s m ⌋ < r. (3)
In a number of experiments listed in [START_REF] Geil | Tables for numbers of zeros with multiplicity at least r[END_REF] we calculated D(i 1 , . . . , i m , r, q, . . . , q) for various choices of m, q and r and for all values of (i 1 , . . . , i m ) satisfying [START_REF] Bras-Amors | Duality for some families of correction capability optimized evaluation codes[END_REF].

We here list the maximal attained improvement obtained by using Theorem 4 rather than using Corollary 1. We do this relatively to the number of points q m . In other words we list in Table 1 the value max i1,...,im

{min{(i 1 + • • • i m )q m-1 /r, q m } -D(i 1 , .
. . , i m , r, q, . . . , q)} /q m for various choices of m, q, r. The experiments also show distinct average improvement. This is illustrated in Table 2 where for fixed q, r, m we list the mean value of

min{(i 1 + • • • + i m )q m-1 , q m } -D(i 1 , . . . , i m , r, q, . . . , q) min{(i 1 + • • • + i m )q m-1 , q m } . ( 4
)
The average is taken over the set of exponents (i 1 , . . . , i m ) = 0 where ⌊i 1 /q⌋ + • • • + ⌊i m /q⌋ < r holds. The values D(i 1 , . . . , i m , r, s 1 , . . . , s m ) may sometimes be time consuming to calculate. Therefore it is relevant to have some closed formula estimates of these numbers. We next present such estimates for the case of two variables. By Remark 1 the following proposition covers all non-trivial cases:

s 2 i 1 r + i 2 r i 1 r -k if (r -k) r r+1 s 1 ≤ i 1 < (r -k)s 1 and 0 ≤ i 2 < ks 2 , (C.2) s 2 i 1 r + ((k + 1)s 2 -i 2 )( i 1 r -k - i 1 r ) + (i 2 -ks 2 )(s 1 - i 1 r ) if (r -k) r r+1 s 1 ≤ i 1 < (r -k)s 1 and ks 2 ≤ i 2 < (k + 1)s 2 , (C.3) s 2 i 1 r + i 2 k + 1 (s 1 - i 1 r ) if (r -k -1)s 1 ≤ i 1 < (r -k) r r+1 s 1 and 0 ≤ i 2 < (k + 1)s 2 . Finally, (C.4) D(i 1 , i 2 , r, s 1 , s 2 ) = s 2 ⌊ i 1 r ⌋ + i 2 (s 1 -⌊ i 1 r ⌋) if s 1 (r -1) ≤ i 1 < s 1 r and 0 ≤ i 2 < s 2 .
The above numbers are at most equal to min{(i

1 s 2 + s 1 i 2 )/r, s 1 s 2 }.
Proof. The estimates are found by treating i 1 and i 2 as rational numbers.

The decoding algorithm

The main ingredient of the algorithm is to find an interpolation polynomial

Q(X 1 , . . . , X m , Z) = Q 0 (X 1 , . . . , X m )+Q 1 (X 1 , . . . , X m )Z+• • •+Q t (X 1 , . . . , X m )Z t
such that Q(X 1 , . . . , X m , F (X 1 , . . . , X m )) cannot have more than n-E different zeros of multiplicity at least r whenever Supp(F ) ⊆ M. The integer E above is the number of errors to be corrected by our list decoding algorithm. To fulfill this requirement we will define appropriate sets of monomials B(i, E, r), i = 1, . . . , t and then require Q i (X 1 , . . . , X m ) to be chosen such that Supp(Q i ) ⊆ B(i, E, r). Rather than using the results from the previous section on all possible choices of F (X 1 , . . . , X m ) with Supp(F ) ⊆ M we only consider the worst cases where the leading monomial of F is contained in the following set:

Definition 4. M = {M ∈ M | if N ∈ M and M |N then M = N }.
Hence, M is so to speak the boarder of M. Definition 5. Given positive integers i, E, r with E < n let

B(i, E, r) = {K ∈ ∆(r, m) | D r (KM i ) < n -E for all M ∈ M}.
Here, D r (X i1 1 , . . . , X im m ) can either be D(i 1 , . . . , i m , r, s 1 , . . . , s m ) or in the case of two variables it can be the numbers from Proposition 3. Another option would be to let D r (X i1 1 , . . . , X im m ) be the number in [START_REF] Augot | A Note on the Generalisation of the Guruswami-Sudan List Decoding Algorithm to Reed-Muller Codes[END_REF]. With a reference to Remark 1 we have defined

∆(r, m) = {X i1 1 • • • X im m | ⌊i 1 /s 1 ⌋ + • • • + ⌊i m /s m ⌋ < r}.
The decoding algorithm calls for positive integers t, E, r such that

t i=1 |B(i, E, r)| > nN (m, r), (5)
where N (m, r) = m+r m+1 is the number of linear equations to be satisfied for a point in F m+1 q to be a zero of Q(X 1 , . . . , X m , Z) of multiplicity at least r. As we will see condition [START_REF] Dvir | Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers[END_REF] ensures that we can correct E errors. We will say that (t, E, r) satisfies the initial condition if given the pair (E, r), t is the smallest integer such that (5) is satisfied. Whenever this is the case we let B ′ (t, E, r) be any subset of B(t, E, r) such that

t-1 i=1 |B(i, E, r)|+|B ′ (t, E, r)| = nN (m, r)+1.
Replacing B(t, E, r) with B ′ (t, E, r) lowers the run time of the algorithm.

Algorithm 1 Input: Received word r = (r 1 , . . . , r n ) ∈ F n q . Set of integers (t, E, r) that satisfies the initial condition and corresponding sets B(1, E, r) . . . , B(t -1, E, r), B ′ (t, E, r).

Step 1 Find non-zero polynomial

Q(X 1 , . . . , X m Z) = Q 0 (X 1 , . . . , X m )+Q 1 (X 1 , . . . , X m )Z+• • •+Q t (X 1 , . . . , X m )Z t such that 1. Supp(Q i ) ⊆ B(i, E, r) for i = 1, . . . , t -1 and Supp(Q t ) ⊆ B ′ (t, EP , r), 2. (P i , r i ) is a zero of Q(X 1 , . . . , X m , Z) of multiplicity at least r, i = 1, . . . , n. Step 2 Find all F (X 1 , . . . , X m ) ∈ F q [X 1 , . . . , X m ] such that (Z -F (X 1 , . . . , X m ))|Q(X 1 , . . . , X m , Z). (6)
Output: A list containing (F (P 1 ), . . . , F (P n )) for all F satisfying (6).

Theorem 5. The output of Algorithm 1 contains all words in E(M, S) within distance E from the received word r. Once the preparation step has been performed the algorithm runs in time O(n 3 ) where n = n m+r m+1 . For given multiplicity r the maximal number of correctable errors E and the corresponding sets B(1, E, r), . . . , B(t-1, E, r), B ′ (t, E, r) can be found in time O(n log(n)r m s ′ |M|/σ) assuming that the values of the function D r are known. Here σ = max{i

1 + • • • + i m |X i1 1 • • • X im m ∈ M} and s ′ = max{s 1 , . . . , s m }.
Proof. The interpolation problem corresponds to n homogeneous linear equations in n + 1 unknowns. Hence, indeed a suitable Q can be found in time O(n 3 ). Now assume Supp(F ) ⊆ M and that dist H (ev S (F ), r) ≤ E. Then P j is a zero of Q(X 1 , . . . , X m , F (X 1 , . . . , X m )) of multiplicity at least r for at least n -E choices of j. By the definition of B(i, E, r) this can, however, only be the case if Q(X 1 , . . . , X m , F (X 1 , . . . , X m )) = 0. Therefore, Z -F (X 1 , . . . , X m ) is a factor in Q(X 1 , . . . , X m , Z). Finding linear factors of polynomials in (F q [X 1 , . . . , X m ])[Z] can be done in time O(n 3 ) by applying Wu's algorithm in [START_REF] Wu | An Algorithm for Finding the Roots of the Polynomials over Order Domains[END_REF] (see [13, p. 20]).

Examples

Example 1. In this example we consider a point ensemble S = S 1 × S 2 with s 1 = 128 and s 2 = 64. We first consider codes E(M, S) where M = {X i1 1 X i2 2 | i 1 + 2i 2 ≤ u}. Note, the weight 2 which corresponds to the number s 1 /s 2 . The performance of Algorithm 1 is independent of the field in which S 1 and S 2 live. In Table 3 we list the number of errors that we can correct when the function D r (i 1 , i 2 ) -is chosen to be D(i 1 , i 2 , r, 80, 80) (column D), -is the closed formula expression from Proposition 3 (column C), -or is the Schwartz-Zippel bound (column S), respectively. The row ⌊(d -1)/2⌋ corresponds to half the minimum distance and the row Dim. is the dimension of the code. We next consider codes where M

= {X i1 1 X i2 2 | i 1 < k 1 , i 2 < k 2 }.
That is, we consider Reed-Solomon product codes. In Table 4 we list the number of errors that can be corrected by Algorithm 1.

Example 2. In this example we consider a point ensemble S = S 1 × S 2 with s 1 = s 2 = 80. We consider codes E(M, S) where M = {X i1 1 X i2 2 | i 1 + i 2 ≤ u} for various values of u. That is, we consider Reed-Muller like codes. The performance of Algorithm 1 is independent on the field in which S 1 and S 2 live. In Table 5 we list the number of errors that we can correct when the function D r (i 1 , i 2 ) -is chosen to be D(i 1 , i 2 , r, 80, 80) (column D), -is the closed formula expression from Proposition 3 (column C), -or is the Schwartz-Zippel bound (column S), respectively. Column A corresponds to a bound from [START_REF] Augot | A Note on the Generalisation of the Guruswami-Sudan List Decoding Algorithm to Reed-Muller Codes[END_REF] on what can be achieved by applying their algorithm. The row A ∞ corresponds to what could theoretically be achieved by the algorithm in [START_REF] Augot | A Note on the Generalisation of the Guruswami-Sudan List Decoding Algorithm to Reed-Muller Codes[END_REF] if one uses high enough multiplicity. Assuming S 1 , S 2 ⊆ F 128 , E(M, S) corresponds to a puncturing of the generalized Reed-Muller code RM 128 (u, 2). This suggest that as an alternative to using Algorithm 1 one could decode with respect to RM 128 (u, 2) treating the punctured points as errors. The best known algorithm to decode RM 128 (u, 2) for the values of u considered in this example is the algorithm by Pellikaan and Wu which uses the subfield subcode approach. Row PW of the table explains what can be achieved by this alternative approach.

Concluding remarks

In this work we presented a decoding algorithm that works for a large class of affine variety codes. By a number of experiments we illustrated the usefulnes of our algorithm. The work was supported in part by Danish Natural Research Council grant 272-07-0266. The authors thank Peter Beelen and Teo Mora for pleasant discussions. Also thanks to L. Grubbe Nielsen for linguistical assistance.

j = 1 ,

 1 . . . , m} and define the affine variety code E(M, S) = Span Fq {ev S (M ) | M ∈ M}.

  Pellikaan and Wu also presented a direct interpretation of the Guruswami-Sudan decoding algorithm for generalized Reed-Muller codes. Even though Augot et al. improved the analysis of the latter algorithm dramatically [1, 2]; the algorithm that uses the subfield subcode is still superior when decoding generalized Reed-Muller codes. The analysis by Augot et al. uses a generalization of the Schwartz-Zippel bound to also deal with multiplicity. A proof of this bound was given by Dvir et al. in [5]. Augot et al. then used their insight to generalize the second algorithm by Pellikaan and Wu (the direct interpretation of the Guruwami-Sudan algorithm) to Reed-Muller like codes over S = S 1 × • • • × S m , when S 1 = • • • = S m , and to Reed-Solomon product codes as well. Based on the generalized Schwartz-Zippel bound they estimated the decoding radius.

Definition 3 .

 3 Let r ∈ N, i 1 , . . . , i m ∈ N 0 . Define D(i 1 , r, s 1 ) = min i1 r , s 1 and for m ≥ 2 D(i 1 , . . . , i m , r, s 1 , . . . , s m ) = max (u1,...,ur)∈A(im,r,sm)

Table 1 .

 1 Maximum improvements relative to q m ; truncated

	m			2				3		4
	r	2	3	4	5	2	3	4	5	2	3
		2 0.25 0.25 0.25 0.25 0.25 0.375 0.375 0.375 0.312 0.375
		3 0.222 0.222 0.222 0.222 0.296 0.296 0.296 0.296 0.296 0.333
		4 0.187 0.187 0.187 0.187 0.281 0.25 0.25 0.265 0.316 0.289
	q	5 0.24 0.16 0.16 0.2	0.256 0.256 0.232 0.24 0.307 0.288
		7 0.204 0.204 0.163 0.142 0.279 0.244 0.227 0.209 0.299 0.276
		8 0.234 0.203 0.171 0.140 0.275 0.25 0.214 0.203 0.299 0.275

Table 2 .

 2 The mean value of (4); truncated For k = 1, . . . , r -1, D(i 1 , i 2 , r, s 1 , s 2 ) is upper bounded by

	m			2				3		4
	r	2	3	4	5	2	3	4	5	2	3
		2 0.363 0.273 0.337 0.291 0.301 0.300 0.342 0.307 0.248 0.260
		3 0.217 0.286 0.228 0.236 0.194 0.224 0.213 0.214 0.158 0.177
		4 0.191 0.197 0.232 0.195 0.158 0.169 0.180 0.172 0.125 0.135
	q	5 0.155 0.167 0.174 0.197 0.139 0.145 0.148 0.153 0.110 0.116
		7 0.128 0.137 0.138 0.138 0.119 0.122 0.121 0.119 0.093 0.098
		8 0.126 0.127 0.134 0.126 0.114 0.115 0.113 0.111 0.089 0.093
	Proposition 3.

(C.1)

Table 3 .

 3 Error correction capabilities for the first codes in Example 1

	u	3			4			7			20
	r D	C	S	D	C	S	D	C	S	D	C	S
	2 5129 5105 4895 4799 4777 4575 4143 4124 3871 2487 2475 2175
	3 5367 5333 5205 5048 5016 4906 4407 4381 4245 2855 2833 2666
	4 5474 5438 5343 5180 5143 5071 4566 4535 4431 3060 3031 2927
	9	5653 5617		5390 5361		4817 4785		3415 3384
	20	5757 5740		5509 5494		4959 4943		3609 3599
	⌊ d-1 2 ⌋ Dim.	3999 6			3967 9			3871 20			3455 121

Table 4 .

 4 Error correction capabilities for the second codes in Example 1.

	(k1, k2)	(4, 7)			(5, 9)			(8, 15)			(21, 41)
	r D	C	S	D	C	S	D	C	S	D	C	S
	2 4036 4015 3519 3655 3639 3071 2820 2808 2111 1061 1055 0
	3 4289 4261 3903 3911 3885 3498 3077 3058 2602 1183 1171 533
	4 4411 4381 4111 4042 4011 3727 3214 3187 2847 1310 1291 831
	9	4598 4487		4244 4124		3455 3313		1567 1365
	20	4711 4662		4364 4310		3588 3526		1704 1615
	⌊ d-1 2 ⌋ Dim.	3720 28			3599 45			3248 120			1935 861

Table 5 .

 5 Error correction capabilities for the codes in Example 2

	u	3				4				7				20
	r D	C	S	A	D	C	S	A	D	C	S	A	D	C	S	A
	2 3594 3571 3399 3310 3317 3297 3119 2999 2693 2679 2479 2302 1279 1279 999 585
	3 3791 3765 3679 3604 3524 3499 3413 3323 2943 2918 2799 2692 1575 1559 1439 1138
	4 3899 3869 3799 3758 3647 3618 3559 3492 3080 3058 2979 2896		1728 1639 1428
	9	4072 4053 4027		3837 3813 3792		3315 3297 3253		2053 2035 1935
	20	4171 4163 4152		3946 3939 3926		3444 3435 3418		2219 2211 2169
	A∞	4257			4042			3558			2368
	PW	3891			3503			2568			0
	⌊ d-1 2 ⌋ Dim.	3079 10			3039 15			2919 36			2399 231