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Transport of Organic Molecules in environment insights

from retardation in sedimentary rocks. Organic molecules are widely studied in environmental sciences, marine chemistry, soil-and geosciences. Among them, anthropogenic organic matter (AOM) can be potentially released from hazardous waste and migrate through geological rock formation or soils. AOM refers to a wide range of compounds, including both ionic and neutral molecules, polar and apolar molecules. The ionic polar molecules are highly soluble in water and more mobile in soils and rocks. The neutral (poly)aromatic compounds are less soluble, often absorbed by soil organic matter, but also more resistent against (bio)degradation. For all these compounds, adsorption or absorption processes can slow down their migration in environmental conditions. Consequently, understanding retardation phenomena of soluble organic matter in rocks and soils is crucial for safety assements of waste storage, decontamination processes, ore extraction and remediation of soils.

Figure 1: Correlation between octanol/water partition coefficient and adsorption of organic molecules. For comparison, data on ions (blue) were divided by content of N.O.M. content of COx clay rock (0.6%).

A considerable effort has been made for 20 years to study migration phenomena in clay-rich geological barriers in the context of radioactive waste storage. Recent studies emphazised the adsorption of organic molecules on clayrocks [START_REF] Rasamimanana | Adsorption of polar organic molecules on sediments: Case-study on Callovian-Oxfordian claystone[END_REF] . One of our main findings is the correlation between adsorption and hydrophilicity of adsorbates, which highlights the role of various minerals as main sorbing phases (Figure 1). Hydrophobic molecules (red dots, log(P APP )>1) are abosrbed by natural organic matter, whereas hydrophilic anions (blue dots, log(P APP )<-1) are mainly adsorbed on oxides or clay minerals. These sorption data are strengthened by various migration experiments, e.g. in-situ diffusion of dissolved gas [START_REF] Vinsot | Natural gas extraction and artificial gas injection experiments in Opalinus Clay, Mont Terri rock laboratory (Switzerland)[END_REF] , or percolation of hydroxyacids in compacted clays [START_REF] Chen | Transport of low molecular weight organic compounds in compacted illite and kaolinite[END_REF] . One of the main remaning issue is the discrepancy between adsorption data measured by batch experiments and retardation factors measured by diffusion experiments on highly compacted rocks [START_REF] Dagnelie | Sorption and diffusion of organic acids through clayrock: Comparison with inorganic anions[END_REF] . Potential origins of these discrepencies, such as the "anionic exclusion" effect, will be disccused. The outcomes will be also discussed on a more general level, dealing with industrial processes and fate of organic molecules in the environment.

  R. Dagnelie, J.-C. Robinet To cite this version: R. Dagnelie, J.-C. Robinet. Transport of Organic Molecules in environment insights from retardation in sedimentary rocks.. Interfaces Against Pollution (IAP2018), Jun 2018, La Grande Motte, France. cea-02339259

6 lactate o-phthalate EDTA Transport of Organic Molecules in environment: insights from retardation in sedimentary rocks.

	R² = 0.77 -6 Octanol/water Partition (Log P app ) -4 -2 0 2 4 R² = 0.77 0 2 4 6 -8 -6 -4 -2 0 2 4 Log K OC -8 Octanol/water Partition (Log P app ) R² = 0.97 6 R² = 0.97 6 8 R² = 0.77 R² = 0.97 0 2 4 6 -8 -6 -4 -2 0 2 4 6 8 Log K OC Octanol/water Partition (Log P app ) R² = 0.77 R² = 0.97 0 2 4 6 -8 -6 -4 -2 0 2 4 6 8 Log K OC Octanol/water Partition (Log P app ) [1] CH 4 C 3 H 8 1 Den-Service d'Etude du Comportement des Radionucléides (SECR), CEA, Université paris-Saclay, F-0 2 4 6 8 Log K OC Carboxylates Chlor. Hydroc. Aromatics Triazines Gas diffusion Hydroxyacids Percolation Carboxylates Chlor. Hydroc. Aromatics Triazines Gas diffusion Hydroxyacids Percolation Carboxylates Chlor. Hydroc. Aromatics Triazines Gas diffusion Hydroxyacids Percolation Carboxylates Chlor. Hydroc. Aromatics Triazines Gas diffusion Hydroxyacids Percolation [2] [3] 91191 Gif-sur-Yvette, France. 2 Andra, R&D Division, parc de la Croix Blanche, 92298, Châtenay-Malabry, France C 2 H Dagnelie Romain V.H. 1 , Robinet J.-C. 2 E-mail contact: romain.dagnelie@cea.fr

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Minjia Shi

Member, IEEE Denis S Krotov
email: krotov@math.nsc.ru

Patrick Solé
email: sole@enst.fr

A New Approach to the Kasami Codes of Type 2

Keywords: Kasami codes, completely regular codes, cyclic

The dual of the Kasami code of length q 2 -1, with 1 q a power of 2, is constructed by concatenating a cyclic MDS 2 code of length q + 1 over Fq with a Simplex code of length 3 q -1. This yields a new derivation of the weight distribution 4 of the Kasami code, a new description of its coset graph, and 5 a new proof that the Kasami code is completely regular. The 6 automorphism groups of the Kasami code and the related q-ary 7 MDS code are determined. New cyclic completely regular codes 8 over finite fields a power of 2, generalized Kasami codes, are 9 constructed; they have coset graphs isomorphic to that of the 10 Kasami codes. Another wide class of completely regular codes, 11 including additive codes, as well as unrestricted codes, is obtained 12 by combining cosets of the Kasami or generalized Kasami code.

13

More generally, we construct generalized Kasami codes of lengths q 2 -1 p-1 , when q is an even power of p, and p itself a power of 2. This is obtained by concatenation of a q-ary cyclic MDS code with a p-ary Simplex code. The resulting code is cyclic with two zeros α, α (p-1)(1+q) , where α denotes a primitive root of F q 2 . When p > 2, we obtain in that way infinitely many new cyclic completely regular codes with a coset graph isomorphic to the coset graph of a Kasami code.

We emphasize that one of the main contributions of our research is in applying the concatenation with the Simplex code to cyclic codes, with the result being cyclic too. In general, the concatenation approach in the construction of linear completely regular codes is known, see [START_REF] Rifà | On the Kronecker product construction of completely transitive q-ary codes[END_REF], [START_REF] Rifà | Completely regular codes with different parameters giving the same distance-regular coset graphs[END_REF], [START_REF] Borges | Completely regular codes by concatenating Hamming codes[END_REF], [START_REF] Borges | On completely regular codes[END_REF]. In particular, in [START_REF] Borges | Completely regular codes by concatenating Hamming codes[END_REF], the concatenation of dual-distance-3 MDS codes with the Simplex code was considered (note that a distance-3 MDS code, even non-linear, is always completely regular [START_REF] Kokkala | On the classification of MDS codes[END_REF]Cor. 6]); the result is a two-weight q-ary code, whose dual is a covering-radius-2 completely regular code. We use a similar approach, but start with a cyclic dualdistance-4 MDS code, and prove also the cyclicity of the resulting p-ary (p is a power of two) completely regular code of covering radius 3, which is known as the Kasami code if p = 2, and is a new cyclic completely regular code if p > 2. In fact, concatenation with the Simplex code can be applied as well to produce a completely regular p-ary code from a linear completely regular q-ary code, q = p k , with the same intersection array, independently on the concrete parameters. Moreover, even more general approach guarantees that from any unrestricted (not necessarily linear or additive) qary completely regular code (and even an equitable partition) we can construct a completely regular code with the same intersection matrix over any alphabet p such that q is a power of p (the alphabet sizes p and q are not required to be prime powers). This approach is based on the existence of a locally bijective homomorphism, known as a covering, see, e.g., [START_REF] Godsil | Algebraic Graph Theory[END_REF]Sect. 6.8], between the Hamming graphs H(n, q) (in the role of the cover) and H( q-1 p-1 n, p) (in the role of the target) of the same degree. With the inverse homomorphism, an equitable partition or completely regular code of the target graph automatically maps to an equitable partition or completely regular code, respectively, of the cover graph, with the same intersection matrix. This combinatorial approach is very powerful and allows to connect equitable partitions of different graphs, especially translation graphs (Cayley graphs over abelian groups), such as, for example, bilinear forms graphs and Hamming graphs. However, it does not always allow to track some nice algebraic properties of the codes such as cyclicity.

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/

The material is arranged as follows. The next section collects the notions and notations needed for the rest of the paper. Section III describes the concatenation approach and the inner code used. Section IV gives a new description of the classical Kasami codes, including their weight distributions, and their automorphism groups. Section V describes the (new) generalized Kasami codes obtained by allowing a more general alphabet in the outer code. This includes a determination of the weight distribution, and of the coset graphs. Section VI provides a combinatorial way to derive new completely regular codes from old, and applies it to the generalized Kasami codes.

Section VII recapitulates the path followed, and points out new directions.

II. DEFINITIONS AND NOTATION

A. Graphs

All graphs in this note are finite, undirected, connected, without loops or multiple edges. In a graph Γ, the neighborhood Γ(v) of the vertex v is the set of vertices connected to v.

The degree of a vertex v is the size of Γ(v). A graph is regular if every vertex has the same degree. The i-neighborhood Γ i (v) is the set of vertices at geodetic distance i to v. A graph is distance regular (DR) if for every pair or vertices ū and v at distance i apart the quantities

b i = |Γ i+1 (ū) ∩ Γ(v)|, c i = |Γ i-1 (ū) ∩ Γ(v)|,
which are referred to as the intersection numbers of the graph, solely depend on i and not on the special choice of the pair (ū, v). The automorphism group of a graph is the set of permutations of the vertices that preserve adjacency.

The Hamming graph H(n, q) (we consider only the case of prime power q) is a distance-regular graph on F n q , two vectors being connected if they are at Hamming distance one. It is well known [START_REF] Markov | On transformations without error propagation, izbrannye trudy (SelectedWorks), vol. II: Teoriya algorifmov i konstruktivnaya matematika[END_REF] that any automorphism of the Hamming graph acts as a permutation π of coordinates and, for every coordinate, a permutation σ i of F q ; i.e., σ • π:

(v 1 , . . . , v n ) → (σ 1 (v π -1 (1) ), . . . , σ n (v π -1 (n) )).

B. Equitable Partitions, Completely Regular Codes

A partition (P 0 , . . . , P k ) of the vertex set of a graph Γ is called an equitable partition (also known as regular partition, partition design, perfect coloring) if there are constants S ij such that

|Γ(v) ∩ P j | = S ij for every v ∈ P i , i,j = 0, . . . , k.
The numbers S ij are referred to as the intersection numbers, and the matrix

(S ij ) k i,j=0
(|C ∩ Γ i (v)|) i=0,

C. Linear and Additive Codes

139

A linear code (that is, a linear subspace of F n q ) of length n, 140 dimension k, minimum distance d over the field F q is called 141 an

[n, k, d] q code. An [n, k, d] q code is called an MDS code 142 if d = n -k + 1.
The duality is understood with respect 143 to the standard inner product. Let A w denote the number of 144 codewords of weight w.

145

Linear codes are partial cases of the additive codes, which 146 are, by definition, the subgroups of the additive group of F n q . A 147 coset of an additive code C is any translate of C by a constant 148 vector. A coset leader is any coset element that minimizes 149 the weight. The weight of a coset is the weight of any of its 150 leaders. The coset graph Γ C of an additive code C is defined 151 on the cosets of C, two cosets being connected if they differ 152 by a coset of weight one.

153 Lemma 1 (see, e.g., [START_REF] Brouwer | Distance-Regular Graphs[END_REF]).

[0, 1, • • • , w, A w , • • • ],
where A w denotes the number of 160 codewords of weight w.

161

D. Cyclic Codes

162

A cyclic code C of length n over F q is, up to polynomial 163 representation of vectors, an ideal of the ring F q [x]/(x n -1). 164 All ideals are principal with a unique monic generator, called 165 the generator polynomial of C, and denoted by g(x). If this 166 polynomial has t irreducible factors over F q , then the code C 167 is called a cyclic code with t zeros. In that case, the dual C ⊥ • k i is the degree of the factor indexed i of g(x),

172

• Tr = Tr q k i /q is the trace from

F q k i down to F q , 173 • a i is arbitrary in F q k i , 174 • α i is a primitive root in F × q k i , 175
In general the relative trace from F q r to F q , where q is a 176 power of a prime, is defined as

177 Tr q r /q (z) = z + z q + • • • + z q r-1 .
If q is prime, then Tr q r /q (z) is the absolute trace. See the 179 theory of the Mattson-Solomon polynomial in [START_REF] Macwilliams | The Theory of Error-Correcting Codes[END_REF] for details 180 and a proof.

181

E. Automorphisms 182

The automorphism group Aut(C) of a q-ary code C of 183 length n is the stabilizer of C in the automorphism group of 184 the Hamming graph H(n, q). Note that every automorphism 185 acts as the composition of a coordinate permutation and a 186 permutation of the alphabet in every coordinate.

We will say that a subgroup A of the automorphism group of the Hamming graph H(n, q) acts imprimitively on the coordinates if the set of coordinates can be partitioned into subsets, called blocks, and denoted by N 1 , …, N t , 1 < t < n, such that the coordinate-permutation part of every automorphism from A sends every block to a block. If there is no such a partition, then the action of the group on the coordinates is primitive (for example, this is the case for Aut(C) of any cyclic code C of prime length).

We denote by Aut0(C) the stabilizer of the all-zero word in Aut(C). The monomial automorphism group AutM(C) of a q-ary code C is defined as the set of all monomial transforms that leave the code wholly invariant [START_REF] Huffman | Fundamentals of Error-Correcting Codes[END_REF]. A monomial transform acting on F n q is the product of a permutation matrix by an invertible diagonal matrix, both in F n×n q . The subgroup consisting of all such transforms with trivial diagonal part is called the permutation part of the automorphism group.

The group AutS(C) consists of the semilinear automorphisms of a q-ary code C, where each semilinear automorphism is the composition τ • μ of a monomial transform μ and an automorphism τ of the field F q (acting simultaneously on the values of all coordinates) such that τ • μ(C) = C. Note that for any linear code C, the groups AutM(C) and AutM(C ⊥ ) are always isomorphic, as well as the groups AutS(C) and AutS(C ⊥ ); the same is not necessarily true for Aut or Aut0.

If q is not prime and F q has a subfield F p , we also use the notation AutL p (C) to denote the subgroup of Aut0(C) consisting of only transformations that are linear over F p . Any automorphism from AutL p (C) acts as the composition of a coordinate permutation and a F p -linear permutation (from GL(log p q, p)) of F q in every coordinate. In particular, if q is a power of 2, then we have AutM(C) ≤ AutS(C) ≤ AutL 2 (C) ≤ Aut0(C) ≤ Aut(C), and if q = 2, then the first four of these groups are the same (the last group, for an additive code, is the product of Aut0(C) and the group of translations by codewords).

III. CONCATENATION WITH THE SIMPLEX CODE

Concatenation is the replacement of the symbols of the codewords of some code, called the outer code, by the codewords of some other code, called the inner code. The role of the inner code in our study is played by a Simplex code; moreover, we mainly focused on the case when this code is cyclic.

Let p be a prime power; let q = p k . Let ξ be a root of unity of order q-1 p-1 in F q . We assume that k and p -1 are coprime; hence, so are q-1 p-1 and p -1 (indeed, by induction,

p k -1 p-1 ≡ k mod p -1)
. It follows that

F × q = bξ l | b ∈ F × p , l ∈ {1, . . . , q-1 p-1 } (1)
and, in particular, all powers of ξ are mutually linear independent over F p . Then, the matrix (ξ 1 , . . . , ξ q-1 p-1 ), considered as a k × q-1 p-1 matrix over F p , is a generator matrix of a Simplex code of size q, where the distance between any two different

codewords is p k-1 [11, Th. 2.7.5].
Encoding the Simplex code. Let L be a nondegenerate 241 F p -linear function from F q to F p . Any such function can be 242 represented as L(•) ≡ Tr q/p (a•) for some a ∈ F q , and in our 243 context we can always assume that L is Tr q/p . Define φ : 244

F q → F q-1 p-1 p by 245 φ(z) = L(zξ 1 ), . . . , L(zξ q-1 p-1 ) ;
246 expand its action to F n q coordinatewise, and to any set of words 247 in F n q wordwise. 248 Lemma 2. For any two words ȳ, z in F n q , we have 249 d(φ(ȳ), φ(z)) = p k-1 d(ȳ, z). I.e., φ is a scaled isometry.

250

Proof: Immediately from the fact that the Simplex code 251 is a one-weight code with nonzero weight p k-1 , we have

252 d(φ(y 1 , . . . , y n ), φ(z 1 , . . . , z n )) 253 = d (φ(y 1 ), . . . , φ(y n )), (φ(z 1 ), . . . , φ(z n )) 254 = n i=1 d(φ(y i ), φ(y i )) = n i=1 p k-1 d(y i , y i ) 255 = p k-1 d((y 1 , . . . , y n ), (z 1 , . . . , z n )). 256 257 Lemma 3. Assume that M = (M ij ) m i=1 n j=1 is an m × n 258 generator matrix of a code C in F n q . Then the matrix 259 Φ(M ) = ⎛ ⎜ ⎝ M 11 ξ 1 ... M 11 ξ q-1 p-1 M 12 ξ 1 . . . M 1n ξ q-1 p-1 ... ... ... ... . . . ... M m1 ξ 1 ... M m1 ξ q-1 p-1 M m2 ξ 1 . . . M mn ξ q-1 p-1 ⎞ ⎟ ⎠, 260 considered as a km × q-1 p-1 n matrix over F p , is a generator 261 matrix of the code φ(C) in F q-1 p-1 n p . 262
Proof: The proof is straightforward. Any codeword c of 263 the code generated by Φ(M ) is a linear combination of its 264 rows, and so is of the form

265 c = m i=1 L i (M i1 ξ 1 ), . . . , L i (M in ξ q-1 p-1 ) 266 for some F p -linear functions L i : F q → F p . Since L i (•) ≡ 267 L(a i •) for some a i ∈ F q , we have 268 c = m i=1 L(a i M i1 ξ 1 ), . . . , L(a i M in ξ q-1 p-1 ) 269 = m i=1 φ(a i M i1 ), . . . , φ(a i M in ) 270 = m i=1 φ(a i M i1 , . . . , a i M in ) 271 = φ m i=1 a i (M i1 , . . . , M in ) . 272
That is, c is a φ-image of some codeword of C.

273

The following lemma provides an alternative explanation 274 to the fact that completely regular codes with identical coset 275 graphs can exist over different alphabets [START_REF] Rifà | Completely regular codes with different parameters giving the same distance-regular coset graphs[END_REF].

Lemma 4. The coset graph Γ C ⊥ of the dual C ⊥ of a linear 277 code C in F n q and the coset graph Γ φ(C) ⊥ of the dual φ(C) ⊥ 278 of its image φ(C) in F q-1 p-1 n p are isomorphic.
(ξ l M 1j , . . . , ξ l M mj ) T , ( 3
)
303 j ∈ {1, . . . , n}, l ∈ 1, . . . , q -1 p -1 , b ∈ F × p , 304
corresponding to the weight-1 words with b in the position

305 (j -1) q-1 p-1 + l. Since {bξ l | b ∈ F × p , l ∈ {1, . . . , q-1 p-1 }} 306 coincides with F × q (see ( 1
1, . . . , q -1 p -1 , . . . , q -1 p -1 (n-1)+1, . . . , q -1 p -1 n .
325

Proof: The action of the monomial automorphism group on a codeword c of C can be represented as a permutation of the coordinates and, for every i from 1 to n the action of an element of GL(m, p) on of the value in the i-th coordinate, understood as an element of F m p . We first consider these two actions separately.

(i) A permutation of coordinates corresponds to a permutation of blocks for φ(C): we see that

φ(z π -1 (1) , . . . , z π -n (n) ) = (φ(z π -1 (1) ), . . . , φ(z π -n (n) ))
is obtained from φ(z 1 , . . . , z n ) by sending each i-th block of coordinates to the place of the φ(i)-th block.

(ii) Next, we show that the action of a F p -linear permutation on the value z i in the i-th coordinate corresponds to a monomial transform inside the i-th block of φ(z 1 , . . . , z n ).

Here, a crucial observation if the following well-known fact.

(*) Any bijective linear transform of the Simplex code is equivalent to the action of some of its monomial automorphisms. This fact is straightforward from the form of a generator matrix of the Simplex code, which consists of a complete collection of pair-wise linearly independent columns of height m. Applying a bijective linear transform to the rows, we obtain another generator matrix with the same property. It is obvious that one such matrix can be obtained from another one by permuting columns and multiplying each column by a nonzero coefficient, that is, by a monomial transform.

Consider the value z i in the i-th position and some permutation τ i of F q that is linear over F p . Since φ is also linear over F p and invertible, we have a bijective linear transform φ(τ i (φ -1 (•))) of the Simplex code. By (*), there is a monomial automorphism σ i of the Simplex code whose action on this code is identical to φ(τ i (φ -1 (•))). Then, we have

φ(τ i (z i )) = σ i (φ(z i )),
i.e., the action of a F p -linear permutation τ i in the i-th coordinate of a word z from F n q corresponds to the action of the monomial transform σ i in the i-th block of coordinates of φ(z).

Summarizing (i) and (ii), we get the following. If the mapping

(z 1 , . . . , z n ) → τ i (z π -1 (1) ), . . . , τ i (z π -n (n) ) belongs to AutL p (C), then (φ(z 1 ), . . . , φ(z n )) → σ i (φ(z π -1 (1) )), . . . , σ i (φ(z π -n (n) ))
maps φ(C) to itself, i.e., belongs to AutM(φ(C)). This proves the statement.

Lemma 6. Assume that C is a linear code over F q and C ⊥ satisfies the following property:

(i) the minimum distance of C ⊥ is at least 4.
Then the groups AutL p (C), AutM(φ(C)), and AutM(φ(C) ⊥ ) are isomorphic.

Proof: We will first deduce from (i) that (ii) Aut0(φ(C) ⊥ ) (and hence, also AutM(φ(C) ⊥ )) acts imprimitively on the coordinates with the blocks

1, . . . , q -1 p -1 , . . . , q -1 p -1 (n-1)+1, . . . , q -1 p -1 n .
(As we see from Lemma 5, this property is indeed necessary, while (i) is only sufficient to guarantee (ii).) To show (ii), we note two other facts. From the code distance ≥ 4 of C ⊥ , we see that (iii) for every codeword of weight 3 in φ(C) ⊥ , the three nonzero coordinates belong to the same block. Indeed, assume that φ(C) ⊥ has a codeword with exactly three nonzero coordinates, with values β, β and β respectively. This means that for some i, i , i , j, j , j such that (i, j) = (i , j ) = (i , j ) = (i, j), the equation

βL(z i ξ j ) + β L(z i ξ j ) + β L(z i ξ j ) = 0
is satisfied for every (z 1 , . . . , z n ) from C. Since L is linear and nondegenerate, this implies

(βξ j )z i + (β ξ j )z i + (β ξ j )z i = 0 for all (z 1 , . . . , z n ) from C.

438

From Lemma 5, we have the inverse correspondence. It 439 remains to note that AutM of a code and its dual are 440 isomorphic.

441

IV. CLASSICAL KASAMI CODES

442

Let q = 2 m for some integer m ≥ 1. Consider the cyclic 443 code M ⊥ q of length q + 1 defined over F q by the check 444 polynomial h(x) = (x -1)(xζ)(xζ q ), for the root 445 ζ = α q-1 of order q + 1 over F q 2 , where α is a primitive 446 root of F q 2 (for some notation reasons, we first define M ⊥ q 447 and then M q as the dual to M ⊥ q ). Note that

ζ q 2 = ζ, so that 448 (x-ζ)(x-ζ q ) = x 2 -(ζ +ζ q )x+ζ q+1 = x 2 -(Tr q 2 /q ζ)x+1 449 has its coefficients in F q . 450 Theorem 1. The code M ⊥ q is MDS of parameters [q+1, 3, q-451 1] q . Its weight distribution is 452 0, 1 , q -1, q 3 -q 2 , q, q 2 -1 , q+1, q 3 -2q 2 + q 2 . 453 (4) 454
Proof: The BCH bound (see e.g. [13, § 7.6]) applied to 455 M q shows that M q , hence M ⊥ q is MDS. The frequency of 456 weight q -1 is derived on applying [13, Corollary 5, p. 320]. 457 A q-1 = (q -1)

q + 1 q -1 = (q -1) q + 1 2 .

458

The frequency of weight q is derived on applying [13, p. 320]. 459

A q = q + 1 3 -2 (q 2 -1)q(q -1) = (q + 1)(q -1).

460

The frequency of weight q + 1 is derived by complementation 461

A q+1 = q 3 -1 -A q-1 -A q .
462 Denote by S q the binary Simplex code of parameters 463 [q -1, m, q 2 ] 2 . The main result of this section is the following 464 theorem.

465

Theorem 2. The concatenation of M ⊥

q with S q is a binary 466 cyclic code K ⊥ q of parameters [q 2 -1, 3m], with nonzeros α, 467 α 1+q , with α a primitive root of F q 2 . 468 Proof: As ζ = α q-1 is a root of order q + 1 of F q 2 , by 469 the facts of Section 2.4, we have 470

M ⊥ q = c + Tr q 2 /q (δζ j ) q+1 j=1 c ∈ F q , δ ∈ F q 2 .
471 Let θ = α 1+q . Clearly θ is a root of order q -1 of F q . The 472 concatenation map φ from F q to S q is conveniently described 473 by using the trace function Tr q/2 . For all β ∈ F q let 474 φ(β) = Tr q/2 (βθ), Tr q/2 (βθ 2 ), . . . , Tr q/2 (βθ q-1 )

475 = Tr q/2 (βθ i ) q-1 i=1 .
Replacing β by the generic coordinate of M ⊥ q , we obtain an 477 expression depending on i and j.

478

Tr q/2 (cθ i ) + Tr q 2 /2 (δα i(q+1)+j(q-1) ).

479 By using the Chinese Remainder Theorem for integers for the 480 product q 2 -1 = (q + 1)× (q -1), we see that the exponent of 481 α in the second sum ranges over 1, . . . , q 2 -1 modulo q 2 -1.

482

This is the Trace formula for a cyclic code with nonzeros α,

483 θ = α 1+q . 484 Corollary 2. The weight distribution of K ⊥ q is 485 0, 1 , q 2 -q 2 , q (q 2 -1) 2 , 486 q 2 2 , q 2 -1 , q 2 + q 2 , q 3 -2q 2 + q 2 .
487 Proof: Concatenation with S q multiplies the weights of 488 M ⊥ q by a factor q 2 , while leaving their frequencies unchanged. 492 Corollary 3. The coset graphs of K q and M q are isomorphic.

493

In particular [START_REF] Borges | Families of nested completely regular codes and distance-regular graphs[END_REF], K q is a completely regular code of diameter 494 3 and intersection array 495 {q 2 -1, q(q -1), 1; 1, q, q 2 -1}.

(

) 5
496

Proof: The first claim holds by Lemma 4, where C = 497 M q , C ⊥ = M ⊥ q , φ(C) = K ⊥ q , and φ(C) ⊥ = K q . Since M q 498 is a completely regular code with intersection array [START_REF] Borges | On completely regular codes[END_REF], see

499

[5, F.5], the coset graphs are distance regular and the code K q 500 is completely regular with the same intersection array.

501

In the rest of this section, we find the automorphism groups 502 of K q and related codes.

503

Remark 2. The concatenation construction for the partial case 504 q = 4 of the code K q was suggested in [4, Sect. 4.1], without 505 considering the cyclicity.

506

A. Automorphism Groups 507

According to Lemma 6, to know the automorphism group 508 of K q , we have to find AutL 2 (M ⊥ q ). We first prove that it 509 coincides with AutS(M ⊥ q ). 510 Lemma 7. The automorphism groups of the linear [q + 1, 3,

511 q -1] q MDS code M ⊥ q satisfy 512 Aut0(M ⊥ q ) = AutL 2 (M ⊥ q ) = AutS(M ⊥ q ).
513

Proof: The proof applies ideas from [START_REF] Gorkunov | The automorphism group of a q-ary Hamming code[END_REF] for differ-514 ent code parameters. Assume that π is a permutation 515 of coordinates and σ = (σ 1 , . . . , σ n ) is a collection 516 of permutations of F q fixing 0 such that σ • π :

517 (v 1 , . . . , v n ) → (σ 1 (v π -1 (1) ), . . . , σ n (v π -1 (n) )) belongs to 518
Aut0(M ⊥ q ). Denote by πM ⊥ q the code whose codewords 519 are obtained from the codewords of M ⊥ q by applying the 520 coordinate permutation π. So, σ(πM ⊥ q ) = M ⊥ q .

521

We split the proof into four steps. The first step is sufficient to see the first equality in the claim of the lemma.

(i) We state that for every coordinate i and every a, b ∈ F q it holds σ

i (a + b) = σ i (a) + σ i (b). Equivalently, σ • π ∈ AutL 2 (M ⊥ q )
. Indeed, consider two codewords ȳ, z ∈ πM ⊥ q of minimum weight, q -1, with zeros in different positions and having a and b in the i-th position. Consider the two words ū = σ(ȳ + z) and v = σ(ȳ) + σ(z). They both lie in M ⊥ q . Moreover, they coincide in every position where ȳ or z has 0. Since ȳ has 0 in two positions and z has 0 in another pair of positions, ū or v differ in less than q -1 positions. As the code distance is q -1, we have ū = v. In particular, their values σ i (a + b) and σ i (a) + σ i (b) in the i-th coordinate coincide, which proves (i).

(ii) We state that for any coordinate i and for any values a and b from F q , it holds

σ 1 (a)σ i (b) = σ 1 (1)σ i (ab). ( 6
)
We first assume that i = 2 and there is a weight-(q -1) codeword ȳ = (1, b, y 3 , . . . , y n ) of πM ⊥ q . Since the codes M ⊥ q and πM ⊥ q are linear and σ • π is an automorphism of M ⊥ q , the words

σ(ȳ) = (σ 1 (1), σ 2 (b), σ 3 (y 3 ), . . . , σ n (y n )) and σ(aȳ) = (σ 1 (a), σ 2 (ab), σ 3 (ay 3 ), . . . , σ n (ay n ))
are codewords of M ⊥ q , as well as the words ū = σ 1 (a)σ(ȳ) and v = σ 1 (1)σ(aȳ). The codewords ū and v both have weight 0 or q -1 and the same nonzero positions; moreover, they coincide in the first coordinate (with the value σ 1 (1)σ 1 (a)). It follows that the distance between ū and v is smaller than the minimum distance of M ⊥ q , and hence ū = v. In particular, the values σ 1 (a)σ 2 (b) and σ 1 (1)σ 2 (ab) in the second position of ū and v coincide, which proves [START_REF] Brouwer | Distance-Regular Graphs[END_REF].

In the next step, we still consider i = 2. A straightforward and well-known property of an MDS code of dimension k (and distance nk + 1) is that there is exactly one codeword having chosen values in chosen k distinct coordinates. By ȳ(l) , l = 2, . . . , n-1, we denote the unique codeword of πM ⊥ q with 1 in the first coordinate and 0 in the l-th and n-th coordinates. For different l and t, the words ȳ(l) and ȳ(t) are different (indeed, ȳ(l) cannot have 0 in the t-th position because its weight cannot be less than the code distance q-1). Hence, they are different in all positions except the first and the last (the distance cannot be less than q -1). So, ȳ(l) , l = 2, . . . , n -1, possess n -2 = q -1 different values in the second position. Therefore, for any given a, (6) holds for q -1 distinct values b ∈ F q . It immediately follows from the bijectivity of σ i that (6) holds for the remaining value of b. So, ( 6) is true for i = 2 for any a and b. Similarly, it is true for any i = 1.

The remaining case i = 1 is derived from the case i = 1.

From [START_REF] Brouwer | Distance-Regular Graphs[END_REF] we see that the ratio σ i (ba) σ i (b) does not depend on b.

By similarity, this holds for any i including i = 1. So, we have

σ 1 (ba) σ 1 (b) = σ 1 (a) σ 1 (1)
, which is ( 6) with i = 1. (i) is proven.

(iii) We state that there is an integer degree d such that

σ i (a) = σ i (1)a d ( 7
)
for every i from 1 to n and every a in F q .

Indeed, assume β is a primitive element. From ( 6) we have 7) by induction on k.

σ i (ab) = σ i (a)(σ 1 (b)/σ 1 (1)). Substituting b = β, a = β k and denoting d = log β (σ 1 (β)/σ 1 (1)), we get σ i (β k+1 ) = σ i (β k ) • β d , which proves (
(iv) The mapping a → a d , where d is from (iii), is an automorphism of the field F q . Indeed, this mapping preserves the multiplication:

(a • b) d = a d • b d
. By (i), it preserves the addition as well.

So, the action of every automorphism σ•π from Aut0(M ⊥ q )

can be represented as the combination of a coordinate permutation, applying an automorphism of F q to the values in all coordinates, and multiplying the values of every coordinate by a constant. By the definition, such automorphism belongs to AutS(M ⊥ q ).

To find AutS(M ⊥ q ), we consider the isomorphic group AutS(M q ). The next two lemmas establish the tight upper and lower bounds for this group, respectively.

Lemma 8. The semilinear automorphism group AutS(M q )

of the code M q coincides with Aut0(M q ) (and, in particular, with the intermediate AutL 2 (M q )) and is isomorphic to a subgroup of the general semilinear group ΓL(2, q).

Proof: Denote by H q the Hamming [q + 1, q -1, 3] q code defined by the generator polynomial h

(x) = (x -ζ)(x -ζ q ).
Note that M q is a distance-4 subcode of H q . From the intersection array ( 5) of the completely regular code M q , we find that the number of vertices at distance at least 3 from

M q is |M q | • q 2 -1 1 • q(q -1) q • 1 q 2 -1 = (q -1)|M q |,
i.e., coincides with |H q \M q |. It follows that H q is a unique distance-3 code of size q q-1 that includes M q . It follows that any automorphism π from Aut0(M q ) also belongs to Aut0(H q ) (otherwise, π(H q ) is another distance-3 code including M q ). It is known that AutS(H q ) is isomorphic to ΓL(2, q) [10, Thm 7.2] and, moreover, Aut0(H q ) = AutS(H q ) [START_REF] Gorkunov | The automorphism group of a q-ary Hamming code[END_REF]. So, all automorphisms from Aut0(M q ) are also semilinear, and we get

Aut0(M q ) = AutS(M q ) ≤ Aut0(H q ) = AutS(H q ) ΓL(2, q).
In [START_REF] Borges | Families of nested completely regular codes and distance-regular graphs[END_REF]Prop. 3.3] (where their C (u) is our K q ), it is shown that Aut0(K q ) contains a subgroup isomorphic to the general linear group GL(2, q). The proof of the following lemma pirates arguments from [START_REF] Borges | Families of nested completely regular codes and distance-regular graphs[END_REF], adopting them to the q-ary code M q , instead of the binary K q , and extending from GL(2, q)

to ΓL(2, q). Lemma 9. The semilinear automorphism group AutS(M q ) of the code M q contains a subgroup of size |ΓL(2, q)|.

Proof: (i) In the first part of the proof, we consider different equations defining M q . For each i, we express ζ i as γ i,0 + γ i,1 α, where γ i,0 , γ i,1 ∈ F q . The code M q consists of all c = (c 1 , . . . , c q+1 ) from F q+1 q such that q+1 i=1

c i = 0, (8)
q+1 i=1 c i ζ i = 0, equivalently, q+1 i=1 c i γ i,j = 0, j = 0, 1. (9) 626
Assuming that (9) holds, we express the left part of (8) as 627 follows.

628

q+1 i=1 c i 2 = q+1 i=1 c 2 i (ζ i ) 0 = q+1 i=1 c 2 i (ζ i )(ζ i ) q 629 = q+1 i=1 c 2 i (γ i,0 + γ i,1 α)(γ i,0 + γ i,1 α) q 630 = q+1 i=1 c 2 i (γ i,0 + γ i,1 α)(γ i,0 + γ i,1 α q ) 631 = q+1 i=1 c 2 i γ 2 i,0 + α q+1 q+1 i=1 c 2 i γ 2 i,1 632 + (α + α q ) q+1 i=1 c 2 i γ i,0 γ i,1 633 = q+1 i=1 c i γ i,0 2 + α q+1 q+1 i=1 c i γ i,1 2 634 + (α + α q ) q+1 i=1 c 2 i γ i,0 γ i,1 635 = (α + α q ) q+1 i=1 c 2 i γ i,0 γ i,1 .
636 Therefore, in the system ( 8)-( 9), the equation ( 8) can be 637 replaced by

638 q+1 i=1 c 2 i γ i,0 γ i,1 = 0. ( 10
) 639
The same arguments can be applied if we replace α by the 640 primitive element α t , where t is a power of two. So, [START_REF] Huffman | Codes and groups[END_REF] can 641 be replaced by

642 q+1 i=1 c 2 i γ (t) i,0 γ (t) i,1 = 0, (11) 643
where γ

(t) i,0 , γ (t)
i,1 ∈ F q are the coefficients in the expansion 644

ζ i = γ (t) i,0 + γ (t) i,1 α t .

645

(ii) Now, consider an arbitrary nonsingular semilinear trans-646 form 647

Ψ : γ 0 + γ 1 α → (aγ 0 + a γ 1 ) + (bγ 0 + b γ 1 )α t 648
of F q 2 as a two-dimensional vector space over F q , where a, a , 649 b, b are coefficients from F q such that det a a b b = ab +a b = 650 0 and t is a power of two. So, (•) t is an automorphism of F q , 651 and Ψ ∈ ΓL(2, q). Denote by ē(i) , i = 1, . . . , q+1, the weight-652 1 vector in F q+1 q with 1 in the i-th coordinate. Consider the 653 mapping ψ that maps ē(i)

to d i ē(ji) , d i ∈ F q if Ψ(ζ i ) = d i ζ ji 654
(by [START_REF] Bannai | Algebraic Combinatorics I: Association Schemes[END_REF], every nonzero element of F q 2 is uniquely represented 655 as dζ l for some l ∈ {1, . . . , q + 1} and d ∈ F × q ). Using the 656 semilinearity identity ψ(λȳ + μz) = λ t ψ(ȳ) + μ t ψ(z), we 657 expand the domain of ψ to the whole vector space F q+1 q . We 658 will show that ψ lies in AutS(M q ). Consider a codeword z = (z 1 , . . . , z q+1 ) ∈ M q . It satisfies 660 [START_REF] Gorkunov | The automorphism group of a q-ary Hamming code[END_REF] and [START_REF] Huffman | Codes and groups[END_REF] with c = z. We have to show that ( 9) and [START_REF] Huffman | Fundamentals of Error-Correcting Codes[END_REF] 661 hold for c = ψ(z). We start with [START_REF] Gorkunov | The automorphism group of a q-ary Hamming code[END_REF]. Note that in the case

662 c = ψ(ē (i) ), the expression q+1 j=1 c j ζ i turns to 663 q+1 j=1 c j ζ j = d i ζ ji = Ψ(ζ i ) 664 = (aγ i,0 + a γ i,1 ) + (bγ i,0 + b γ i,1 )α t .
665 So, for

666 c = ψ(z) = ψ q+1 i=1 z i ē(i) = q+1 i=1 z t i ψ(ē (i) ) 667 we have 668 q+1 j=1 c j ζ j = q+1 i=1 Ψ(z i ζ i ) 669 = q+1 i=1 z t i (aγ i,0 + a γ i,1 ) + (bγ i,0 + b γ i,1 )α t 670 = (a + bα) q+1 i=1 z i γ i,0 + (a + b α) q+1 i=1 z i γ i,1 t = 0

671

(the last two sums are zeros because of ( 9) for c = z), which 672 is exactly [START_REF] Gorkunov | The automorphism group of a q-ary Hamming code[END_REF].

673

Next, deal with [START_REF] Huffman | Fundamentals of Error-Correcting Codes[END_REF]. If c = ψ(ē (i) ), then the only nonzero 674 element of c is the j i -th element, with the value c ji = d i , and

675 d i ζ ji = c ji γ (t) ji,0 + c ji γ (t)
ji,1 α t = (aγ i,0 + a γ i,1 ) t + (bγ i,0 + 676 b γ i,1 ) t α t . Hence, for c = ψ(ē (i) ) we have

677 q+1 j=1 c j γ (t) j,0 • c j γ (t)
j,1 = (aγ i,0 + a γ i,1 ) t (bγ i,0 + b γ i,1 ) t .

678

Therefore, for c = ψ(z) we have

679 q+1 j=1 c j γ (t) j,0 • c j γ (t) j,1 680 = q+1 i=1 z t i (aγ i,0 + a γ i,1 ) t • z t i (bγ i,0 + b γ i,1 ) t 681 = a t b t q+1 i=1 z i γ i,0 2t + a t b t q+1 i=1 z i γ i,1 2t 682 + (ab + a b) t q+1 i=1 z 2 i γ i,0 γ i,1 t = 0

683

(the last three sums are zeros because of ( 9) and [START_REF] Huffman | Codes and groups[END_REF] for 684 c = z), which is exactly [START_REF] Huffman | Fundamentals of Error-Correcting Codes[END_REF]. Clearly, different Ψ correspond 685 to different ψ; so, the number of semilinear automorphisms of

686

M q is at least |ΓL(2, q)|. AutS of the codes M ⊥ q , M q , K q , K ⊥ q are isomorphic to 691 each other and to ΓL(2, q). The groups AutM(M ⊥ q ) and 692

AutM(M q ) are isomorphic to GL(2, q).

693

Proof: For the binary codes, we have Aut0(K q ) = AutL 2 (K q ) = AutS(K q ) = AutM(K q ) and Aut0(K ⊥ q ) = AutL 2 (K ⊥ q ) = AutS(K ⊥ q ) = AutM(K ⊥ q ). By Lemma 7, AutS(M ⊥ q ) = AutL 2 (M ⊥ q ) = Aut0(M ⊥ q ). By Lemmas 8 and 9, AutS(M q ) = AutL 2 (M q ) = Aut0(M q ) ∼ = ΓL (2, q). By Lemma 6, AutM(K ⊥ q ) ∼ = AutL 2 (M ⊥ q ). For dual codes we have AutS(K q ) = AutS(K ⊥ q ), AutS(M ⊥ q ) = AutS(M q ), and AutM(M ⊥ q ) = AutM(M q ). By the arguments in the proof of Lemma 8, the subgroup AutM(M q ) of AutS(M q ) coincides with the subgroup AutM(H q ) of AutS(H q ) and so isomorphic to GL(2, q).

V. GENERALIZED KASAMI CODES

Let q = p m for some integer m ≥ 1, and some power of two p. Thus the preceding section is the special case p = 2 of the present section. As in the previous section, we consider the cyclic code M ⊥ q of length q + 1 defined over F q by the check polynomial h(x) = (x -1)(xζ)(xζ q ), for some root ζ of order q + 1 over F q 2 . By Theorem 1, M ⊥ q is a [q + 1, 3, q -1] q MDS code with weight distribution (4).

Denote by S q the p-ary Simplex code of parameters [ q-1 p-1 , m, q p ] p . This code is cyclic when m and p -1 are coprime. The main result of this section is the following theorem.

Theorem 4. The concatenation of M ⊥

q with S q is a p- 1) , with α a primitive root of F q 2 . If m and p -1 are coprime, then this code is cyclic.

ary code K p q ⊥ of parameters [ q 2 -1 p-1 , 3m] p , with nonzeros α, α (1+q)(p-
Proof: The proof is analogous to that of Theorem 2. The only difference is the definition of θ = α (1+q)(p-1) , which is now a root of order q-1 p-1 in F q 2 . When p > 2, as far as we know, the codes K p q are new, and will be called generalized Kasami codes in this paper.

Corollary 4. The weight distribution of K

p q ⊥ is 0, 1 , q 2 -q p , q (q 2 -1) 2 , q 2 p , q 2 -1 , q 2 + q p , q 3 -2q 2 + q 2 .
Proof: Concatenation with S q multiplies the weights of M ⊥ q by a factor q p , while leaving their frequencies unchanged.

Corollary 5. The coset graphs of K p q and M q are identical. In particular, when p is even, K p q is completely regular of diameter 3, and intersection array {q 2 -1, q(q -1), 1; 1, q, q 2 -1}.

Proof: This follows by Corollary 1.

Remark 3. The monomial automorphism group AutM(K p q ) of the generalized Kasami code is isomorphic to ΓL(2, q), by arguments similar to Theorem 3. We conjecture that it coinsides with AutS(K p q ) and Aut0(K p q ), because it is so for M q = K q q and K q = K 2 q . However, our current tools do not allow to prove this easily. 0) , C (1) , C (2) , C (3) ) is an equitable partition 760 of H(

⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ 0 p 2m -1 0 0 1 q -2 p 2m -q 0 0 q p 2m -q -2 1 0 0 p 2m -1 0 ⎞ ⎟ ⎟ ⎠ . 762 (12)
763 By standard double-counting arguments, we have 764 |C (3) |/|C (0) | = |C (2) |/|C (1) | = (p 2mq)/q = r -1, where 765 r = p 2m /q. Let C 0 = C (0) = C and let {C 1 , . . . , C r-1 } be 766 a partition of C (3) into cosets of C (0) . Each of the cosets of 767 C is a completely regular code with the same parameters.

768

For each i from 0 to r -1, by C r+i we denote the set of 769 vertices at distance 1 from C i . In particular, C (1) = C r and 770

C (2) = ∪ r-1 i=1 C r+i .

771

We state that (C i ) 2r-1 i=0 is an equitable partition with the

772 quotient matrix 773 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 • • • 0 p 2m -1 0 • • • 0 0 0 • • • 0 0 p 2m -1 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • 0 0 0 • • • p 2m -1 1 0 • • • 0 q-2 q • • • q 0 1 • • • 0 q q-2 • • • q . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • 1 q q • • • q-2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 774
To show this, we first note that it is a partition; this is 784

Now, it is straightforward to check that denoting

B (0) = k-1 i=0 C i , B (3) = r-1 i=k C i , B (1) = k-1 i=0 C r+i , B (2) = r-1 i=k C r+i ,
we obtain an equitable partition (B (0) , B (1) , B (2) , B (3) ) with the intersection matrix

⎛ ⎜ ⎜ ⎝ 0 p 2m -1 0 0 1 kq -2 p 2m -kq 0 0 kq p 2m -kq -2 1 0 0 p 2m -1 0 ⎞ ⎟ ⎟ ⎠ .
Therefore, the set B k = B (0) is a completely regular code with the required intersection array.

(ii) We first note that the set B 2 constructed as in (i) is automatically additive. Then, we see that B 2 satisfies the hypothesis of the proposition for the code C, with 2q instead of q. So, repeating the construction log 2 k times, we get an additive completely regular code with the required parameters.

In the case m = 1, the code C from Proposition 1 can be chosen as a 1 p -th part of the p-ary Hamming code of length p + 1 in the sense of [5, (F.5)] (note that p is a power of 2), in particular, as the code M q , in notation of Section IV.

In [START_REF] Borges | On completely regular codes[END_REF], this code is classified as belonging to the group of completely regular codes obtained from perfect codes. Proposition 1 shows that not only 1 p -th part, but also 1 2 i -th part, i = 1, 2, . . . , log 2 p, of the p-ary Hamming code of length p+1 can be chosen to be an additive completely regular code. This remark essentially extends the class of completely regular codes obtained from perfect codes.

VII. CONCLUSION AND OPEN PROBLEMS

In this article, we have used a concatenation scheme to produce new completely regular codes from known ones. The use of a one-weight code as the inner code is essential in preserving the coset graph structure from the outer code to concatenated code, as evidenced in the proof of Lemma 4. The Simplex code is essentially the only one-weight code in the Hamming graph by a classical result of Bonisoli [START_REF] Bonisoli | Every equidistant linear code is a sequence of dual Hamming codes[END_REF]. It is worth investigating systematically if another choice of the outer code than the cyclic MDS code used here can lead to new constructions of completely regular codes. More generally, replacing the Hamming graph by other distance regular graphs with a larger choice of one-weight codes could lead to new constructions of completely regular codes in these graphs.

  1,2,... of C with respect to a vertex v depends only on the distance from v to C.

168

  admits a trace representation with t terms of the form 169 c(a 1 , . . . , a t ) = t i=1Tr(a i α j i )

279Proof:

  We first observe that the number of cosets of C ⊥ 280 coincides with size of the code C, which is the same as the size 281 of the code φ(C), which is, in its turn, the number of cosets of 282 φ(C) ⊥ . That is, the considered graphs have the same number 283 of vertices.284Let C be a code over F q with a generator matrix M , which 285 is a check matrix of the dual code C ⊥ . The cosets of C ⊥ 286 naturally correspond to the syndromes under the action of 287 the check matrix M . By the definition of the coset graph 288 Γ C ⊥ , two cosets are adjacent in the coset graph if and only 289 if the difference between the corresponding syndromes is the 290 syndrome of a weight-1 word. We call the syndromes of the 291 weight-1 words the connecting syndromes because they play 292 a connecting role for the coset graph (this terminology is 293 in agree with that for a more general class of graphs called 294 Cayley graphs). For the check matrix M = (M ij ) m i=1 n j=1 over 295 F q , the connecting syndromes are 296 a(M 1j , . . . , M mj ) T , j ∈ {1, . . . , n}, a ∈ F × q (2) 297 (corresponding to the weight-1 words with a in the j-th 298 position). 299 Now, consider the matrix Φ(M ), which, by Lemma 3, is a 300 generator matrix for φ(C) and a check matrix for φ(C) ⊥ . The 301 connecting syndromes are 302 b

  The last equation contradicts to the code distance of C ⊥ , unless i = i = i , i.e., the three coordinates are from the same block.On the other hand, because the dual of a Simplex code is a 1-perfect Hamming code, we have (iv) every two coordinates from the same block are nonzero coordinates for some weight-3 codeword in φ(C) ⊥ . Indeed, in the generator matrix of the Simplex code, the sum of any two columns is proportional to some other column. The same relation takes place between the corresponding coordinates from the same block of the concatenated code, which implies the required property.From (iii) and (iv), we see that the code properties of φ(C) ⊥ distinguish the pairs of coordinates in the same block and the pairs of coordinates in different blocks. Hence, the automorphism group does not break blocks and we have (ii).Next, we observe that property (i) excludes some degenerate cases with vanishing coordinates: (v) every codeword of the Simplex code occurs in every block, over the codewords of the concatenated code φ(C). Indeed, this obviously equivalent to the fact that for every symbol of F q and every position, there is a codeword of C with the given symbol in the given position. Since C is linear, it is the same as to say that there is no vanishing coordinate in C. Equivalently, C ⊥ does not contain a weight-1 codeword, which follows immediately from (i). Now, we have that every automorphism from AutM(φ(C) ⊥ ) permutes blocks, permutes coordinates inside each block, and permutes the values of F p for every coordinate (fixing 0). The same is true for AutM(φ(C)), because it is isomorphic to AutM(φ(C) ⊥ ) with the isomorphism keeping the permutation part. According to (v), the last two steps (permuting coordinates inside each block, and permuting the values of F p for every coordinate) must be compatible with the automorphism group of the Simplex code, and result in a permutation of its codewords.A permutation of the codewords of the Simplex code in each block, for the code φ(C), corresponds to a permutation of the elements of F q in the corresponding coordinate for C; and a permutation of the blocks corresponds to a permutation of the coordinates for C. So, we see that every automorphism 433 from AutM(φ(C)) corresponds to some automorphism of C: 434 AutM(φ(C)) Aut0(C). 435 Moreover, because all considered transformations are linear 436 over F p , we have 437 AutM(φ(C)) AutL p (C).

Remark 1 .

 1 This result is derived in [13, Th. 32, p. 252] by 491 the complex formula of [13, Ch. 6, Th. 2].

687Theorem 3 .

 3 Summarizing the results above, we find the automorphism 688 groups of the Kasami and related codes.689 All the automorphism groups Aut0, AutL 2 , 690

775

  straightforward from the fact that C 0 , …, C r-1 are at distance 776 3 from each other (from s 33 = 0 and s 23 = 1 in (12)). Next, 777 we see that all the neighbors of C i , i = 0, . . . , r -1 are in 778 C r+i . Finally, every vertex of C r+i has one neighbor from C i 779 (by the definition), exactly s 11 = q -2 neighbors from C r+i 780 (because C i is completely regular with intersection array (12)), 781 and exactly s 21 = q neighbors from C r+j for every j from 782 {0, . . . , r -1}\{i} (because C j is completely regular with 783 intersection array (12)).

899

  Patrick Solé received the Ingénieur and Docteur-Ingénieur degrees both 900 from Ecole Nationale Supérieure des Télécommunications, Paris, France, 901 in 1984 and 1987, respectively, and the habilitation à diriger des recherches 902 from Université de Nice-Sophia Antipolis, Sophia Antipolis, France, in 1993. 903 He has held visiting positions in Syracuse University, Syracuse, NY, from 904 1987 to 1989, Macquarie University, Sydney, Australia, from 1994 to 1996, 905 and Lille University, Lille, France, from 1999 to 2000. Since 1989, he has 906 been a permanent member of the CNRS and became Directeur de Recherche 907 in 1996. His research interests include coding theory (codes over rings, 908 algebraic codes), interconnection networks (graph spectra, expanders), vector 909 quantization (lattices), and cryptography (boolean functions, pseudo random 910 sequences). He is the author of over 150 journal papers and four books, and 911 co-recipient of the best paper award for Information Theory in 1995, for The 912 Z 4 -linearity of Kerdock, Preparata, Goethals and related codes, a reference 913 quoted more than 1300 times as per Google Scholar. He was associate editor 914 of the Transactions from 1996 till 1999. He has been associate editor of 915 Advances in Mathematics of Communication since 2007.

  An additive code with dis-154 tance at least 3 is completely regular with intersec-155 tion array {b 0 , . . . , b ρ-1 ; c 1 , . . . , c ρ } if and only if the 156 coset graph is distance-regular with intersection numbers 157 b 0 , . . . , b ρ-1 , c 1 , . . . , c ρ .

158

The weight distribution of a code is displayed as 159

  VI. UNION OF COSETS743In this section, we prove a simple but important fact that 744 cosets of a linear completely regular code from the classes 745 considered in this paper can be merged to result in another 746 completely regular code with different parameters, which can 747 be linear, additive, or even non-additive. Let p, q be powers of 2, and let C be an

	749
	750	additive completely regular code over F p with intersection
	751	array {p 2m -1, p 2m -q, 1; 1, q, p 2m -1}. Then
	752 753	(i) for every k from 1 to p 2m q -1, there is a completely regular code B k with intersection array
	754	I k = {p 2m -1, p 2m -kq, 1; 1, kq, p 2m -1},
	755	B k being the union of k cosets of C;
	756	(ii) moreover, if k is a power of 2, then there is an additive
		code with intersection array I k .

748 Proposition 1. 757 Proof: (i) Denote by C (d) the set of vertices at distance 758 d from C, d = 0, 1, 2, 3. By the definition of a completely 759 regular code, (C

  00 s 01 s 02 s 03 s 10 s 11 s 12 s 13 s 20 s 21 s 22 s 23 s 30 s 31 s 32 s 33

	761	⎛ ⎝ ⎜ ⎜	s	p 2m -1 p-1 , p) with the intersection matrix

  of China in 2010. From August 2012 to August 2013, he was a visiting 879 researcher with School of Physical & Mathematical Sciences, Nanyang 880 Technological University, Singapore. In July-August 2016, he was a visiting 881 researcher with Telecom Paris Tech, Paris, France. He has been teaching at 882 the School of Mathematical Sciences of Anhui University since 2007. Since 883 April 2012, he has been the Masters supervisor and Associate Professor with 884 the School of Mathematical Sciences, Anhui University of China. He has 885 been supervisor of Doctor since 2014. Since 2017, he has been Professor 886 with the School of Mathematical Sciences, Anhui University of China. He is 887 the author of over 80 journal papers and two books. His research interests 888 include algebraic coding theory and cryptography. Krotov received the Bachelor's degree in Mathematics in 1995 and 890 the Master's degree in 1997, both from Novosibirsk State University, the Ph.D. 891 and Dr.Sc. degrees in Discrete Mathematics and Theoretical Cybernetics from 892 Sobolev Institute of Mathematics, Novosibirsk, in 2000 and 2011, respectively. Since 1997, he has been with Theoretical Cybernetics Department, Sobolev 894 Institute of Mathematics, where he is currently a Chief Researcher. In June-895 August 2003, he was a visiting researcher with Pohang University of Science 896 and Technology, Korea. For four months in 2018 and 2019, he visited the 897 Anhui University, China. His research interest includes subjects related to 898 algebraic combinatorics, coding theory, and graph theory.

889

Denis S.

893

ACKNOWLEDGMENT

The authors thanks Evgeny Gorkunov for useful discussions and the anonymous referees for valuable suggestions towards improving the paper.

This work was supported in part by the National Natural Science Foundation of China under Grant 61672036, in part by the Excellent Youth Foundation of Natural Science Foundation of Anhui Province under Grant 1808085J20, in part by the Academic Fund for Outstanding Talents in Universities under Grant gxbjZD03, and in part by the Program of Fundamental Scientific Researches of the Siberian Branch of the Russian Academy of Sciences under Project 0314-2019-0016.

M. Shi is with the Key Laboratory of