Minjia Shi

Michael Kiermaier

Sascha Kurz

Patrick Solé

Three-weight codes over rings and strongly walk regular graphs *

Keywords: strongly walk-regular graphs, three-weight codes, homogeneous weight, Kerdock codes, Teichmüller codes Primary 05 E 30, Secondary 94 B 05

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over Z p m , for p a prime, and more generally over chain rings of depth m, and with a residue field of size q, a prime power. In the case of p = m = 2, strong necessary conditions ( diophantine equations) on the weight distribution are derived, leading to a partial classification in modest length. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.

Introduction

Since the seminal article of Delsarte [START_REF] Delsarte | Weights of linear codes and strongly regular normed spaces[END_REF], there is a well-known interplay between two-weight codes and strongly regular graphs (SRG) via the coset graph of the dual code [START_REF] Brouwer | Distance-regular graphs[END_REF][START_REF] Brouwer | Spectra of graphs[END_REF]. Strongly walk-regular graphs (SWRG) were introduced in [START_REF] Van Dam | Strongly walk-regular graphs[END_REF] as a generalization of strongly regular graphs. Instead of a regularity condition bearing on rings. Section 3 derives the preliminary results coming from spectral graph theory. Section 4 focuses on Z 4 codes, derives strong necessary conditions, gives examples in short lengths, as well as an infinite family related to the Kerdock codes. Section 5 describes how the main results extend when the alphabet is a finite chain ring, and gives an infinite family related to generalized Teichmüller codes.

Background 2.1 Homogeneous weight

For simplicity, let R be a finite commutative ring. A linear code C over R of length n is an R-submodule of the standard R-module R n . The dual code of C is defined as

C ⊥ = {x ∈ R n | x • C = 0}
, where x • y = x 1 y 1 + • • • + x n y n denotes the standard inner product on R n . Definition 1. Let R be a finite ring. A function w : R → R is called a homogeneous weight, if w(0) = 0 and 1. if Rx = Ry then w(x) = w(y) for all x, y ∈ R;

2. there exists a real number γ = 0 such that y∈Rx w(y) = γ|Rx| for all x ∈ R \ {0}.

It is known from the work in [START_REF] Constantinescu | A metric for codes over residue class rings[END_REF][START_REF] Greferath | Finite-ring combinatorics and MacWilliams' equivalence theorem[END_REF][START_REF] Honold | Weighted modules and representations of codes[END_REF]] that a homogeneous weight exists for all finite commutative rings R, and is uniquely determined up to the normalization factor γ. In the sequel we will denote the homogeneous weight by w hom , assuming that γ has been fixed. As usual we extend w hom to a weight function on R n by w hom (x 1 , . . . , x n ) = n i=1 w hom (x i ).

The set R n then forms a distance space under the homogeneous distance d hom defined by d hom (x, y) = w hom (x -y). A code C ⊆ R n is called a (homogeneous) three-weight code with weights w 1 < w 2 < w 3 if every non-zero codeword has homogeneous weight w 1 , w 2 , or w 3 , and the three weights actually occur. If the code C is linear then on C × C the homogeneous distance takes only these three values and zero.

A matrix G ∈ R ×n is called a generator matrix of the linear code C if the rows of G generate C as an R-module. 1. regular if {x • g i | x ∈ R } = R for i = 1, . . . , n; 2. projective if Rg i = Rg j for any pair of distinct coordinates i, j ∈ {1, . . . , n}.

In particular, a code is regular if every column contains at least one unit element. We extend this notion also to a single column vector.

In the sequel we shall be interested mostly in the case R = Z p m , the ring of integers modulo a prime power p m . In this case the homogeneous weight is given by

w hom (x) =          0 if x = 0, p m-1 if 0 = x ∈ p m-1 Z p m , (p -1)p m-2 otherwise.
The chosen normalization constant is γ = (p -1)p m-2 , which is the most convenient for our purposes. For m ≥ 2 the homogeneous weight is always an integer. For general p and m, regular and projective is characterized by a metric property of the dual [START_REF] Krotov | Z 2k dual binary codes[END_REF].

For p = m = 2, the homogeneous distance coincide with the classical Lee distance of [START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF]. For a Z 4 -code to be both regular and projective is equivalent to d ⊥ ≥ 3, where d ⊥ denotes the dual Lee distance.

Triple sum and s-sum sets

We extend the definition of triple sum sets given in [START_REF] Courteau | On triple sum sets and two or three-weight codes[END_REF] from finite fields to finite commutative rings R with a unit group R × . The set Ω ⊆ R k is a triple sum set (TSS) if it is stable by scalar multiplication by units ∈ R × and if there are integer constants σ 0 and σ 1 such that a non-zero h ∈ R k can be written as

h = x + y + z, with x, y, z ∈ Ω, • σ 0 times if h ∈ Ω, • σ 1 times if h ∈ R k \ Ω.
This can generalized immediately to the notion of an s-sum set by replacing the

equation h = x + y + z by h = s i=1
x i where each x i ∈ Ω.

Syndrome graph

An eigenvalue of a graph Γ (i.e., an eigenvalue of its adjacency matrix) is called a restricted eigenvalue if there is a corresponding eigenvector which is not a multiple of the all-one vector 1. Note that for an η-regular connected graph, the restricted eigenvalues are simply the eigenvalues different from η. Definition 3. Let T be a finite abelian group and S ⊆ T a subset satisfying S = -S and 0 T / ∈ S. The corresponding Cayley graph C(T, S) has vertex set equal to T ; two vertices g, h ∈ T are adjacent in C(T, S) iff g -h ∈ S.

The conditions imposed on S ensure that C(T, S) is a simple graph. The graph C(T, S) is regular of degree |S|, and it is connected iff S generates the group T . We use the same idea as in [START_REF] Calderbank | The geometry of two-weight codes[END_REF] to associate with any linear code C over Z p m a certain Cayley graph, called the syndrome graph of C and denoted by Γ(C).

Let H = [h 1 | • • • | h n ] ∈ (Z p m ) ×n be a parity-check matrix of C. The vertex set of Γ(C) is V = Hx; x ∈ (Z p m ) n ,
the column space of H, which is isomorphic to the dual code C ⊥ . The elements of V are called syndromes of C. Two syndromes Hx, Hy are adjacent in Γ(C) if they differ by a unit-multiple of a column of H :

Hx ∼ Hy : ⇐⇒ H(x -y) = uh i for some 1 ≤ i ≤ n and u ∈ Z × p m . (1)
It is obvious that Γ(C) is the Cayley graph of V corresponding to the generating S =

{uh i ; u ∈ Z × p m , 1 ≤ i ≤ n}. If C is regular then 0 / ∈ S and -S ⊆ S, implying that Γ(C) is simple. Moreover, if C is projective, Γ(C) is regular of degree |S| = p m-1 (p -1)n and has p mn |C| vertices. The latter follows from |C| C ⊥ = |(Z p m ) n | = p mn .
As defined, Γ(C) depends on the particular choice of H. The terminology "syndrome graph of C" is justified, however, since Γ(C) has another representation as a "coset graph" of C, which shows that its isomorphism type is determined by C. It is well-known that there is a one-to-one correspondence between syndromes of a code and its cosets [START_REF] Ling | Coding Theory A First Course[END_REF], given by x + C → Hx. The condition in (1) can be restated as H(x -y -ue i ) = 0, where e i denotes the ith standard unit vector in (Z p m ) n . Thus Γ(C) is isomorphic to the graph whose vertices are the cosets of C and in which two vertices x + C, y + C are adjacent if their difference is a coset of the form ue i + C. This is the Cayley graph of the quotient group (Z p m ) n /C with generating set

S * = ∪ n i=1 ∪ u∈Z × p m (ue i + C).
Now, we recall the relation between the weight distribution of a linear code over Z p m and the eigenvalues of the syndrome graph of its dual code. This extension of Lemma 3.4 in [START_REF] Calderbank | The geometry of two-weight codes[END_REF] was derived in [START_REF] Shi | The Geometry of Two-Weight Codes over Z p m[END_REF] Theorem 4.1 and its proof is omitted here.

Theorem 1. Suppose that C is a regular, projective linear code over Z p m with homogeneous weights w i and corresponding weight distribution

A i = |{x ∈ C; w hom (x) = w i }|.
Then the eigenvalues of Γ(C ⊥ ) are n(p -1)p m-1 -pw i with multiplicity A i .

The coset graph with b loops Γ(C) b of a code C is obtained from Γ(C) by adding b loops around every vertex. The following result is immediate from the previous theorem. The proof is omitted. (In general, we call dual weight distribution of a code the weight distribution of the dual code).

Corollary 1. If C is a Z p m -code, with dual weight distribution [ w i , A i ], such that C ⊥ is regular and projective, then the spectrum of Γ(C) b is {(b + n(p -1)p m-1 -pw i ) A i }.
Thus A i is the frequency of the homogeneous weight w i in C ⊥ and the multiplicity of the eigenvalue b + n(p -1)p m-1 -pw i .

Spectral graph theory

Let R k denote the set of all regular vectors in R k , i.e.,

R k = ω ∈ R k : {ω • x : x ∈ R k } = R .
If Ω ⊆ R k , we denote by C(Ω) the regular and projective code of length n = |Ω| |R × | obtained as the kernel of the k × n matrix H with columns the projectively nonequivalent elements of Ω. Thus H is the check matrix of C(Ω). To simplify notation, let Γ C(Ω) = Γ(Ω) and Γ b C(Ω) = Γ(Ω) b , where we also write Γ(Ω) 0 for Γ(Ω). The following result is immediate from the definitions in the introduction, and will be used repeatedly and implicitly in the rest of the paper. Theorem 2. Keep the above notation. Assume that C(Ω) is regular and projective. The following are equivalent

• Ω (resp Ω ∪ 0) is an s-sum set,

• Γ(Ω) (resp. Γ(Ω) 1 ) is an s-SWRG.

Proof. Observe that Γ(Ω) is a Cayley graph on the group (R k , +) with generators the columns of H. Thus the definition of an s-sum set can be regarded as a statement on the number of paths of length s between 0 and a linear combination of s columns of H. (By translation, we can assume that 0 is the first endpoint of the path).

The following result comes from the well-known connection between SRGs and two-weight codes. It is described for codes over fields in [START_REF] Brouwer | Distance-regular graphs[END_REF][START_REF] Brouwer | Spectra of graphs[END_REF], and for codes over rings in [START_REF] Byrne | Ring geometries, two-weight codes and strongly regular graphs[END_REF].

Theorem 3. If C(Ω) ⊥ is a two-weight code, then both Ω ∪ 0 and Ω are s-sum sets for all s > 1.

Proof. If C(Ω) ⊥ is a two-weight code, it is well-known that Γ(Ω) is an SRG [START_REF] Byrne | Ring geometries, two-weight codes and strongly regular graphs[END_REF]. The result follows by Theorem 2, and the observation, made in the introduction, that SRGs are s-SWRG for all s > 1.

The following result was observed in [START_REF] Griera | On s-sum sets and projective codes[END_REF] Theorem III.6 for the Hamming weight. Theorem 4. If Ω or Ω ∪ 0 is an s-sum set, then C(Ω) ⊥ has at most three nonzero weights. Furthermore, if s is even, then C(Ω) ⊥ is a 2-weight code.

Proof. This is a consequence of [START_REF] Van Dam | Strongly walk-regular graphs[END_REF]Theorem 3.4] applied to the the graph Γ(Ω), and based on Theorem 2.

We come to the coding-theoretic characterization of 3-sum sets which is contained for finite fields in [START_REF] Courteau | On triple sum sets and two or three-weight codes[END_REF]Th. 2.1]. The consideration of syndrome graphs with loops might seem artificial but is justified by the existence of several non-trivial examples, as evidenced by Section 4.

The relation with the 3-SWRG property and the eigenvalues of a connected regular graph was obtained in [START_REF] Van Dam | Strongly walk-regular graphs[END_REF] Proposition 4.1:

Theorem 5. Let Γ be a connected regular graph with four distinct eigenvalues k > θ 1 > θ 2 > θ 3 . Then Γ is a 3-SWRG iff θ 1 + θ 2 + θ 3 = 0.
In our setting the formulas for the eigenvalues in Theorem 1 and Corollary 1 give: Theorem 6. Assume that C(Ω) ⊥ is of length n and has three nonzero weights w 1 < w 2 < w 3 . Let b be any non-negative integer.

• Ω is a TSS iff w 1 + w 2 + w 3 = 3n(p -1)p m-2 ; Note that if m ≥ 2 and p = 3, then the second case of Theorem 6 is impossible. More generally, for the third case of Theorem 6 the assumptions m ≥ 2 and p = 3 imply that b has to be divisible by p. Then, the condition may be rewritten as We will see in Subsection 4.2 that none of these possibilities can be attained by a Z 4 -code and the smallest possible length for p = m = 2 and b = 0 is indeed n = 6. Proposition 4.3 in [START_REF] Van Dam | Strongly walk-regular graphs[END_REF] gives an explicit diophantine equation bearing on the eigenvalues of Γ to be an s-SWRG:

• Ω ∪ 0 is a TSS iff w 1 + w 2 + w 3 = 3 1+n(p-1)p m-1 p ; • Γ b C(Ω) is a 3-SWRG iff w 1 + w 2 + w 3 = 3 b+n(p-1)p m-
w 1 + w 2 + w 3 ≥ 3n(p -1)p m-2 and w 1 + w 2 + w 3 ≡ 0 (mod 3).
Theorem 7. Let Γ be a connected regular graph with four distinct eigenvalues k > θ 1 > θ 2 > θ 3 and s ≥ 3. Then Γ is an s-SWRG iff (θ 2 -θ 3 )θ s 1 + (θ 3 -θ 1 )θ s 2 + (θ 1 -θ 2 )θ s 3 = 0. ( 2
)
We remark that for s = 3 Equation ( 2) Theorem 5 is contained as a special case. Similar as in the proof of Theorem 6, we can use the formulas for the eigenvalues in Theorem 1 and Corollary 1 to transfer [START_REF] Van Dam | Strongly walk-regular graphs[END_REF]Proposition 4.3] to s-sum sets: Theorem 8. Assume that C(Ω) ⊥ is of length n and has three nonzero weights w 1 < w 2 < w 3 . Let b ≥ 0 and s ≥ 3 be integers. Note that w 2 = 12 = n in all cases. This can be partially explained by the fact that Equation (2) admits the solution θ 2 = 0, θ 1 = -θ 3 for all odd s. In other words, a graph Γ with these eigenvalues is an s-SWRG for all odd s > 1, cf. [START_REF] Van Dam | Strongly walk-regular graphs[END_REF]Proposition 4.2]. Using Theorem 2 and Theorem 1, this observation translates into a result that is new in the context of s-sum sets: Theorem 9. Assume that C(Ω) ⊥ is of length n and has three nonzero weights w 1 < w 2 = n(p -1)p m-2 < w 3 , with w 1 + w 3 = 2n(p -1)p m-2 . Then Ω is an s-sum set for all odd s > 1.

is equivalent to (θ 1 -θ 2 )(θ 1 -θ 3 )(θ 2 - θ 3 )(θ 1 + θ 2 + θ 3 ) = 0, i.e.,
• Ω is an s-sum set iff (w 3 -w 2 )(n(p -1)p m-2 -pw 1 ) s + (w 1 -w 3 )(n(p -1)p m-2 - pw 2 ) s + (w 2 -w 1 )(n(p -1)p m-2 -pw 3 ) s = 0. • Ω ∪ 0 is an s-sum set iff (w 3 -w 2 )(1 + n(p -1)p m-2 -pw 1 ) s + (w 1 -w 3 )(1 + n(p -1)p m-2 -pw 2 ) s + (w 2 -w 1 )(1 + n(p -1)p m-2 -pw 2 ) s = 0. • Γ b C(Ω) is an s-SWRG iff (w 3 -w 2 )(b + n(p -1)p m-2 -pw 1 ) s + (w 1 -w 3 )(b + n(p -1)p m-2 -pw 2 ) s + (w 2 -w 1 )(b + n(p -1)p m-2 -pw 2 ) s = 0.
In [START_REF] Van Dam | Strongly walk-regular graphs[END_REF]Theorem 4.4] it is essentially proven that if Equation (2) admits a solution (θ 1 , θ 2 , θ 3 ) with θ 1 > θ 2 > θ 3 for two different values of s > 1, then θ 2 = 0 and θ 1 = -θ 3 . This directly leads to: Theorem 10. Assume that C(Ω) ⊥ is of length n and has three nonzero weights

w 1 < w 2 < w 3 . If w 2 = n(p -1)p m-2 , or w 1 + w 3 = 2n(p -1)p m-2 , then there is at most one s > 1 such that Ω is an s-sum set.
If we are not in the situation of Theorem 9, then any given code can yield an s-sum set for at most one value of s. There are other solutions of Equation ( 2), e.g.

(5, 1 2 (-3 + √ 281), 1 2 (-3 - √ 281
)) for s = 5. However, the found solutions are nonintegral and no graph that is an s-SWRG and attains these eigenvalues is known, see [11, p. 808]. Since the eigenvalues corresponding to an s-sum set are integral, we state: Conjecture 1. Ω is an s-sum set for some s > 3 iff

(w 1 , w 2 , w 3 ) = (w 1 , n(p -1)p m-2 , 2n(p -1)p m-2 -w 1 ), for some 0 < w 1 < n(p -1)p m-2 .
At least for the special case p = m = 2, i.e., Z 4 -codes, Conjecture 1 seems to be true also for s = 3, see Conjecture 2 in Section 4. In [START_REF] Coma | All s-sum sets of type 2 or 3 are triple-sum sets[END_REF] it has been shown that all s-sum sets of a special type also have to be 3-sum sets.

Z -codes

Here we consider Z 4 linear codes and assume that the reader is familiar with [START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF]. When we say that C is an [n, k] code, where k = (k 1 , k 2 ) is the shape, this means that C is a linear code over Z 4 of length n, which admits a generator matrix of the form

G = I k 1 A B 0 2I k 2 2D , (3)
where I k 1 and I k 2 denote the k 1 × k 1 and k 2 × k 2 identity matrices, respectively, A and D are 0, 1 valued matrices , and B is a Z 4 matrix. We assume that the stated generator matrix consists of n columns (and k 1 + k 2 rows). We will say that the length n is effective to mean that the generator matrix does not contain an allzero vector 0 as a column. The code size, i.e., the number of codewords, is given by #

C = 4 k 1 • 2 k 2 = 2 2k 1 +k 2 .
For Z 4 -codes we note that the homogeneous weight coincides with the Lee weight.

We note the following compatibility between a Z 4 -code and its Gray image.

w 1 + w 2 + w 3 = 3 b + n(p -1)p m-1 p = 3 b + 2n 2 ,
which is the same with p = 2, m = 1 and length 2n.

By A i we denote the number of codewords of C of homogeneous weight i and by B i we denote the number of codewords of homogeneous weight i of its dual code C ⊥ . The cardinality of the dual code is #C ⊥ = 2 2n-2k 1 -k 2 . Since we assume that a corresponding generator matrix does not have a zero column, we have B 1 = 0 in our context. As mentioned at the end of Subsection 2.1, every Z 4 code has minimum dual distance d ⊥ ≥ 3 iff it is regular and projective. The numbers A i can be encoded in a weight enumerator of C:

Hom C (X, Y ) = c∈C X 2n-w hom (c) Y w hom (c) = 2n i=0 A i X 2n-w i Y w i . ( 4
)
The corresponding MacWilliams identity is given by Hom [START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF]. Note that this formula equals the Hamming weight enumerator for a linear code over F 2 with cardinality #C = 2 k if we replace n by 2n and k by 2k 1 + k 2 . As mentioned in the introduction and used in the proof of Lemma 1, the same conclusion can be obtained via the Gray map. Thus, we can easily rewrite the classical MacWilliams identities [START_REF] Macwilliams | The theory of error-CorrectingCodes[END_REF] or the Pless power moments [START_REF] Pless | Power moment identities on weight distributions in error correcting codes[END_REF] for the Hamming weight over F 2 . Using A 0 = B 0 = 1 and B 1 = 0 the first four (Pless) power moments for the w hom are given by:

C (X, Y ) = 1 #C ⊥ •Hom C ⊥ (X+ Y, X -Y ), see e.g.
i>0 A i = 2 2k 1 +k 2 -1, ( 5
)
i≥0 iA i = 2 2k 1 +k 2 n, ( 6
) i≥0 i 2 A i = 2 2k 1 +k 2 -1 (B 2 + n(2n + 1)), ( 7
) i≥0 i 3 A i = 2 2k 1 +k 2 -2 (3(B 2 n -B 3 ) + 2n 2 (2n + 3)). (8)
We remark, that the first three equations (including B 1 ) can also be found in [START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF]Theorem 3.1].

In this section we are interested in structural results for l-weight codes where l is small. Mostly we will restrict our attention to projective codes only. In some cases we will require additional constraints on the sum l i=1 w i of weights (which are motivated by strongly walk regular graphs, see e.g. [START_REF] Van Dam | Strongly walk-regular graphs[END_REF] and see also [START_REF] Shi | Three weight codes, triple sum sets and strongly walk regular graphs[END_REF], which studies the field case).

Theoretical results for Z 4 -codes with few weights

In this section we give only very few results that are of interest for the remaining part of this paper. For more information we refer the interested reader to e.g. [START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF][START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF], where one-Lee weight codes and projective two-Lee weight codes are studied. For every pair (k 1 , k 2 ) of non-negative integers there exists exactly one non-isomorphic 1-weight [n, (k 1 , k 2 )] code (for a unique length n depending on k 1 and k 2 ), see [START_REF] Carlet | One-weight Z 4 -linear codes, Coding Theory, Cryptography and Related Areas[END_REF].

Lemma 1. Let C be an [n, (k 1 , k 2 )] code. By C 2 we denote the subcode of C spanned by the codewords of even weight. The cardinality of C 2 is either 2 2k 1 +k 2 -1 or 2 2k 1 +k 2 and all codewords of C 2 have an even weight.

Proof. Let E be the code consisting of codewords of C of even Lee weight. This code is linear as the dual of the code spanned by the all-2 vector, so that C 2 = E ∩ C. Thus C 2 is linear and its size depends on whether 2 • 1 ∈ C ⊥ or not.

We also call C 2 the even-weight subcode of C. Lemma 2. Let C be an [n, (k 1 , k 2 )] code with t ≥ 2 different nonzero weights. Then, at most t -1 weights can be odd.

Proof. Let c ∈ C be a codeword of odd weight, then 2c ∈ C is a codeword of even weight.

Lemma 3. Let C be an [n, (k 1 , k 2 )] three-weight code with d ⊥ ≥ 3 and weights w 1 , w 2 , and w 3 that occur A 1 , A 2 , and A 3 times, respectively. Using the abbreviation y = 2 2k 1 +k 2 -1 we have

A 1 = y • (2n 2 -2nw 2 -2nw 3 + 2w 2 w 3 + n) -w 2 w 3 (w 2 -w 1 )(w 3 -w 1 ) (9)
A 2 = y • (2n 2 -2nw 1 -2nw 3 + 2w 1 w 3 + n) -w 1 w 3 (w 2 -w 3 )(w 2 -w 1 ) (10)
A 3 = y • (2n 2 -2nw 1 -2nw 2 + 2w 1 w 2 + n) -w 1 w 2 (w 3 -w 1 )(w 3 -w 2 ) ( 11
)
and

3B 3 = 2n 2 (2n + 3) -(w 1 + w 2 + w 3 ) 2n(2n + 1) -4w 1 w 2 w 3 +2 (w 1 w 2 + w 1 w 3 + w 2 w 3 ) 2n + 2w 1 w 2 w 3 y
for the number of dual codewords of weight 3.

Proof. Solving the first three power moment equations ( 5)-( 7) for A 1 , A 2 , and A 3 gives the first statement. Plugging into the fourth power moment equation ( 8) and solving for 3B 3 gives the last equation of the statement.

We remark that the integrality of B 3 implies that the product of the three weights w 1 w 2 w 3 is divisible by y/2 = 2 2k1 -k2 -2 . Now let us look at the first case of Theorem 6, which is w 1 + w 2 + w 3 = 3n in our situation. Our first observation is that the length n has to be even. Proof. Assume that n is odd. Then, also w 1 + w 2 + w 3 = 3n is odd. Using Lemma 2 we conclude that exactly one weight -say w 2 -is odd. We consider the even weight subcode C 2 of C. Due to Lemma 1 C 2 only contains codewords of even weight and has cardinality #C 2 = 2 2k 1 +k 2 -1 , i.e., A 2 = 2 2k 1 +k 2 -1 . Since C is projective C 2 has effective length n ∈ {n -1, n}. Equation ( 6) gives

w 1 A 1 + w 2 A 2 + w 3 A 3 = 2 2k 1 +k 2 n for C and w 1 A 1 + w 3 A 3 = 2 2k 1 +k 2 -1 n for C 2 .
Taking the difference and dividing by A 2 = 2 2k 1 +k 2 -1 yields w 2 = n + (n -n ). Since w 2 , n are odd and n -n ∈ {0, 1}, we have n = n and w 2 = n. So, let us write w 1 = n -t and w 3 = n + t for some odd integer t (observe that w 1 and w 3 have to be even). With this, Equation (10) and A 2 = 2 2k 1 +k 2 -1 = y gives (y -1)t 2 = (y -n)n. Note that t = 1 gives y = n + 1 or n = 1. In the first case we have A 1 = 0 and w 1 = 0 in the second, so that we can assume t ≥ 3. Solving the above equations for n gives

n = y 2 ± -4t 2 y + 4t 2 + y 2 2 .
So, we consider the diophantine equation

x 2 = -4t 2 y + 4t 2 + y 2 , ( 12
)
where y is a power of 2 and all integers are positive. 1 In our situation we have t ≥ 3, so that y ≥ 64 = 2 6 , since x 2 would be negative for t ≥ 3 and y ≤ 32. Now let us set

x = 2z and y = 2 r+1 , i.e., r ≥ 5. With this we can rewrite Equation ( 12) to

(z -t)(z + t) = 2 2r -2 r+1 t 2 . ( 13
)
Since t is odd, z has to be odd too. We conclude that either z -t or z + t has to be divisible by 2 r from the fact that the greatest common divisor of z -t and z + t divides 2t and r ≥ 5.

Assume that z = s2 r -t for some nonzero integer s (this is the case that 2 r divides z+t). Plugging in into Equation ( 13), dividing by 2 r+1 , and simplifying gives

(2 r-1 s -t) • s = 2 r-1 -t 2 , so that 2 r-1 • s 2 -1 = t • (s -t).
Note that s 2 ≥ 1. If s 2 = 1, then s = t or t = 0, which contradicts t ≥ 3. Thus, the left hand side is positive and s > t ≥ 3. This implies t(s -t) ≤ s(s -t) < 2 r-1 (s 2 -1) -contradiction.

Assume that z = s2 r + t for some nonzero integer s (this is the case that 2 r divides z-t). Plugging in into Equation ( 13), dividing by 2 r+1 , and simplifying gives

(2 r-1 s + t) • s = 2 r-1 -t 2 , so that 2 r-1 • s 2 -1 = t • (-s -t).
Note that s 2 ≥ 1. If s 2 = 1, then s = -t or t = 0, which contradicts t ≥ 3. Thus, the left hand side is positive and -s > t ≥ 3. This implies t(s-t) ≤ s(s-t) < 2 r-1 (s 2 -1) -contradiction.

We remark that in the case where also C 2 is projective the proof can be shortened significantly since [START_REF] Byrne | Properties of codes with two homogeneous weights[END_REF]Corollary 16] implies that 2t is a power of two. An example where Lemma 4 excludes parameters of a code is given by n = 29, 2k 1 + k 2 = 8, w = [START_REF] Huffman | Fundamentals of error correcting codes[END_REF][START_REF] Krotov | Z 2k dual binary codes[END_REF][START_REF] Ma | A survey of partial differences sets[END_REF], and weight distribution 0 1 24 76 31 128 32 51 . Applying the MacWilliams transform gives B 0 = 1, B 1 = B 2 = 0, B 3 = 164, and indeed all B i are non-negative integers. Moreover, the number of even-weight codewords is 128, i.e., half the size of the code.

Next we show that under the assumption of Theorem 10 the length and the three weights all are divisible by large powers of 2 if the length is suitably large. Proposition 2. Let C be an [n, (k 1 , k 2 )] 3-weight code with d ⊥ ≥ 3 and weights satisfying w 1 + w 2 + w 3 = 3n and w 2 = n. For each positive integer r there exists an integer N (r) such that n ≥ N (r) implies that 2 r divides n and 2 r-1 divides all three weights w 1 , w 2 , w 3 .

Proof. Let t be a positive integer with w 1 = n -t and w 3 = n + t. With this (and y = 2 2k 1 +k 2 -1 ) the equations of Lemma 3 are equivalent to

A 1 = n(y -n -t) 2t 2 (14)
A 2 = t 2 (2y -1) -n(y -n) t 2 (15)
A 3 = n(y -n + t) 2t 2 (16)
and

3B 3 = 2n(n -t)(n + t) y . ( 17
)
Since A 3 -A 1 = n t the effective length n has to be divisible by t. From A 2 ∈ N we conclude that t 2 divides n(n -y). So, if p l divides t for some odd prime p, then p 2l has to divide n since y is a power of 2. As an abbreviation we set k = 2k 1 + k 2 . Now let us try to parameterize t = 2 u • v and n = 2 x • v 2 • z for odd positive integers v, z and non-negative integers u, x. Plugging in and simplifying gives

A 1 = z • (2 k-u-1 -2 x-u v 2 z -v) 2 u+1-x (18)
A 2 = 2 2(x-u) v 2 z 2 + 2 k -2 x+k-2u-1 z -1 ( 19
)
A 3 = z • (2 k-u-1 -2 x-u v 2 z + v) 2 u+1-x (20) 3B 3 = v 4 z • (2 x-u vz -1) • (2 x-u vz + 1) 2 k-x-2u-2 , ( 21
)
where u ≤ x (t divides n) and

x ≤ k -1 (n ≤ 2 k -1 since C is projective). If k -x -2u -2 ≥ 1 then B 3 ∈ N and v, z ≡ 1 (mod 2) imply u = x.
Since gcd(vz -1, vz + 1) = 2, we have that 2 k-3u-3 either divides vz -1 or vz + 1. So, we use the parameterization vz = s • 2 k-3u-3 + α for some positive integer s and

α ∈ {-1, 1}. With this A 1 > 0 gives 2 k-u-1 -v s • 2 k-3u-3 + α + 1 > 0, so that vs < 2 2u+2 , i.e., sv ≤ 2 2u+2 -1. Now A 2 > 0 gives v 2 z 2 + 2 k > 2 k-u-1 z, which is equivalent to sv(vz) 2 + 2 k sv > s2 k-u-1 vz = s 2 2 2k-4u-4 + αs2 k-u-1 > s 2 2 2k-4u-4 -s2 k+2u+2 . (22)
Since sv ≤ 2 2u+2 -1 the left hand side is at most

s 2 2 2k-4u-4 -s 2 2 2k-6u-6 + sα2 k-u -sα2 k-3u-2 +α 2 2 2u+2 -α 2 + 2 k+2u+2 -2 k ≤ s 2 2 2k-4u-4 -s 2 2 2k-6u-6 + 2s2 k-u + 2 2u+2 + 2 k+2u+2 ≤ s 2 2 2k-4u-4 -s 2 2 2k-6u-6 + 3s2 k+2u+2 . Thus 4s • 2 k+2u+2 > s 2 2 2k-6u-6 (23)
has to be satisfied, so that k ≤ 8u + 9 and

x = u ≥ k -9 8 . ( 24
)
Otherwise we have k ≤ x + 2u + 2. From A 1 > 0 we conclude y -n > 0. Since y -n is an integer and both y and n are divisible by 2 x we have y -n ≥ 2 x . Now A 2 > 0, Equation ( 15), and 2y

-1 ≤ 2 k imply v 2 2 2u • 2 k -2 x v 2 z • 2 x > 0, so that k + 2u > 2x, i.e., k ≥ 2x -2u + 1. ( 25
)
Combined with k ≤ x + 2u + 2 we obtain x ≤ 4u + 1 and k ≤ 6u + 3, i.e.,

x ≥ u ≥ k -3 6 . (26)
In both cases we can conclude x ≥ u ≥ k-9 8 , so that the result follows from n < 2 k . Conjecture 2. Let C be an [n, (k 1 , k 2 )] 3-weight code with d ⊥ ≥ 3 and weights satisfying w 1 + w 2 + w 3 = 3n and w 1 < w 2 < w 3 . Then, w 2 = n.

We remark that in order to prove Conjecture 2 the Diophantine conditions of Lemma 3, i.e., A 1 , A 2 , A 3 ∈ N >0 and B 3 ∈ N ≥0 are not sufficient. Up to n = 50 we have the following exceptional tuples (n, w 1 , w 2 , w 3 , y, A 1 , A 2 , A 3 , B 3 ):

• [START_REF] Krawtchouk | Sur une généralisation des polynomes d'Hermite[END_REF][START_REF] Huffman | Fundamentals of error correcting codes[END_REF][START_REF] Krotov | Z 2k dual binary codes[END_REF][START_REF] Ma | A survey of partial differences sets[END_REF]64,76,128,51,164) • (33, [START_REF] Krawtchouk | Sur une généralisation des polynomes d'Hermite[END_REF][START_REF] Ma | A survey of partial differences sets[END_REF][START_REF] Shi | Two and three weight codes over F p + uF p[END_REF]64,64,111,80,157) • [START_REF] Macwilliams | The theory of error-CorrectingCodes[END_REF][START_REF] Ling | Coding Theory A First Course[END_REF][START_REF] Ma | A survey of partial differences sets[END_REF][START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF]128,64,299,148,[START_REF] Shi | Codes and Rings: theory and practice[END_REF] • (50, 46, 48, 56, 64, 32, 145, 78, 580) So, from Lemma 4 we can conclude that Conjecture 2 is true for all n < 34. We remark that the examples for n ∈ {29, 34, 50} satisfy B i ∈ N ≥0 for all 1 ≤ i ≤ n, i.e., the MacWilliams identities are not sufficient in order to conclude the non-existence of the corresponding code.

Classification of 3-weights Z 4 -codes with d ⊥ ≥ 3 and short length

Having Theorem 6 in mind we aim to classify 3-weights Z 4 -codes with d ⊥ ≥ 3 and short length. Optimal linear codes over Z 4 with length n ≤ 7 were classified in [START_REF] Wong | Classification of small optimal linear codes over Z 4[END_REF]. Without the restriction to optimal codes, all linear codes over Z 4 with length n ≤ 10 and 2k 1 + k 2 ≤ 8 were classified in [START_REF] Feulner | Canonization of linear codes over Z 4[END_REF] for the homogeneous weight. The corresponding tables at http://www.algorithm.uni-bayreuth.de/en/research/ Coding_Theory/CanonicalForm/Classification/index.html contain the complete counts for n ≤ 10. E.g. for n = 10 and (k 1 , k 2 ) = (5, 0) there are exactly 8 848 026 non-isomorphic codes. For each given n our approach is to loop over all 1 ≤ w 1 < w 2 < w 3 ≤ 2n such that A 1 , A 2 , A 3 , B 3 , as specified in Lemma 3, are integers with A 1 , A 2 , A 3 ≥ 1 and B 3 ≥ 0. Since 2 2k 1 +k 2 -2 has to divide w 1 w 2 w 3 , which is due to B 3 ∈ N, we obtain a finite list of possibilities for each length n. Note that d ⊥ ≥ 3 implies k 1 ≥ 1. With respect to Theorem 6 we consider all remaining tuples where w 1 + w 2 + w 3 ≥ 3n and w 1 + w 2 + w 3 ≡ 0 (mod 3). In Table 1 we list the parameters that can be attained by a Z 4 codes and refer to a corresponding generator matrix. As abbreviations we use w = (w 1 , w 2 , w 3 ), S = w 1 + w 2 + w 3 , and A = (A 1 , A 2 , A 3 ). The parameters of computationally excluded codes are listed in Table 2 and Table 3. For the latter we perform an exhaustive search using the given restrictions on the weights w i , the length n, and the shape (k 1 , k 2 ). We remark that more examples can be found for the parameters:

G 1 6,1 =    101112 010331 002200    , G 1 6,2 =    100122 010212 001331    , G 1 6,3 =      101112 011121 002002 000220      n w A (k 1 , k 2 ) → G 6 (4,6,8) (6,16,9) (2, 1) → G 1 6,1 6 (4,6,8) (18,24,21) (3
, 0) → G 1 6,2 , (2, 2) → G 1 6,3 8 (4,8,12) (1,27,3) (2, 1) → G 1 8,1 8 (4,8,12) (5,51,7) (3, 0
) → G 1 8,2 , (2, 2) → G 1 8,3 3 (2,4,6) (15,15,1) (2, 1) → G 2 3,1 5 (4,6,8) (16,12,3) (2
G 1 8,1 =    10101122 01112311 00220202    , G 1 8,2 =    10001222 01022111 00110013    , G 1 8,3 = 
• n = 15, k = 9, w = [START_REF] Delsarte | Weights of linear codes and strongly regular normed spaces[END_REF][START_REF] Goethals | Nonlinear codes defined by quadratic forms over GF (2)[END_REF][START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF], A = (190, 255, 66);

• n = 18, k = 8, w = [START_REF] Goethals | Nonlinear codes defined by quadratic forms over GF (2)[END_REF][START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF][START_REF] Huffman | Fundamentals of error correcting codes[END_REF], A = (153, 72, 30);

• n = 22, k = 7, w = (20, 24, 28), A = (71, [START_REF] Wong | Classification of small optimal linear codes over Z 4[END_REF][START_REF] Ding | Linear codes from some 2-designs[END_REF].

n 2k 1 + k 2 w A (k 1 , k 2 ) 2 3 (1,2,3) (1,3,3) (1,

Cyclic Kerdock code

The Kerdock code K of length 2 s with s ≥ 3 odd over Z 4 can be expressed as

K = Z 4 j ⊕ Q,
where j denotes the all-one vector. The code Q is a free code of size 4 s with weight spectrum 2 s , 2 s ± 2 s-1 2 .

By puncturing on the first coordinate we obtain a cyclic code of length n = 2 s -1 that was used in [START_REF] Solé | A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties[END_REF] to generate low correlation quadriphase sequences. This code is called K -in [START_REF] Hammons | The Z 4 -Linearity of Kerdock, Preparata, Goethals and Related Codes[END_REF] and has the same weights as Q. It can be shown to be projective by the same type of arguments as in [20, §3.3]. The sum of these weights being 3 • 2 s = 3(n + 1), we see that, by Theorem 6 with b = 2, the syndrome graph of K - with a double loop at each vertex is a 3-SWRG. It can be easily checked that the parameters of K -go in line with our results the previous subsection.

We observe that the binary images have a length congruent to 2 (mod 4), which is enough to show they are not linear, since by [27, Corollary 1], we should have a length multiple of 4. This is an alternative proof that Kerdock codes cannot be linear. The classical proof builds on the fact that the Preparata code, or any code with the same weight distribution is nonlinear [START_REF] Goethals | Nonlinear codes defined by quadratic forms over GF (2)[END_REF].

Extension to chain rings

In this section, we briefly indicate how the preceding work extends when replacing Z m p by a finite chain ring R of depth r ≥ 2, say, and residue field F q , for some prime power q. Recall that such a ring is a local ring whose lattice of ideals form a chain of length r. In particular, the maximal ideal is principal, generated by an element γ, a nilpotent element of nilpotency index r [START_REF] Shi | Codes and Rings: theory and practice[END_REF]Chap. 3]. An example is Z m p , when r = m, and q = p. The definition of the homogeneous weight becomes

w hom (x) =          0 if x = 0, q r-1 if 0 = x ∈ (γ e-1
), (q -1)q r-2 otherwise.

The definition of the syndrome graph in Subsection 2.3 to this generalized situation only needs the obvious modifications. Also the proof of [START_REF] Shi | The Geometry of Two-Weight Codes over Z p m[END_REF]Theorem 4.1] can be generalized to conclude the connection between coset graph eigenvalues and weights: Theorem 11. Suppose that C is a regular, projective linear code over R with homogeneous weights w i and corresponding weight distribution

A i = |{x ∈ C; w hom (x) = w i }|.
Then the eigenvalues of Γ(C ⊥ ) are n(q -1)q r-1 -qw i with multiplicity A i .

In particular, for the coset graph with loops, we have the following result.

Corollary 2. If C is an R-code, of minimum distance at least three, with dual weight distribution [ w i , A i ], then the spectrum of Γ(C) b is {(b + n(q -1)q r-1 -qw i ) A i }.
Thus A i is the frequency of the homogeneous weight w i in C ⊥ and the multiplicity of the eigenvalue b + n(q -1)q r-1 -qw i .

From there, upon observing that Theorem 2 is valid for any finite commutative ring, the main condition for the existence of a TSS becomes Theorem 12. Assume that C(Ω) ⊥ is of length n and has three nonzero weights w 1 < w 2 < w 3 . Let b be any integer.

• Ω is a TSS iff w 1 + w 2 + w 3 = 3n(q -1)q r-2 ;

• Ω ∪ 0 is a TSS iff w 1 + w 2 + w 3 = 3 1+n(q-1)q r-1 q ;

• Γ b C(Ω) is a 3-SWRG iff w 1 + w 2 + w 3 = 3 b+n(q-1)q r-1 q ;

We remark that r ≥ 2 and q not being divisible by 3 the second case is impossible and b has to be divisible by q in the third case.

Trace codes over chain rings

Consider the chain ring F p + uF p , with u 2 = 0. This is a chain ring with r = 2 and q = p. The codes C(m, p) of length p 2m -p m 2 with m singly even in [38, §5.1] are defined as

{(T r(ax)) x∈L | a ∈ R},
where L = Q + uF m p , Q is the set of squares of F p m , and R = F p m + uF p m . Note that, since m is even, we have F × p ⊆ Q. This implies that these codes are replications by a factor p -1 of projective codes P (m, p) say of length n = p 2m -p m 2(p-1) . These latter codes have weights

p 2m-1 -p m-1 , p 2m-1 -p m-1 -p m-1 (p m/2 + 1), p 2m-1 -p m-1 + p m-1 (p m/2 -1),
and so satisfy the relation

w 1 + w 2 + w 3 = 3(1 -1/p)n,
for the weight of the Gray map in [START_REF] Shi | Two and three weight codes over F p + uF p[END_REF], normalized by a factor 2. This shows, by the first condition of Theorem 12, that the coset graph Γ(C) is a 3-SWRG, or, equivalently, that the Ω such that C(Ω) = C is a TSS.

Generalized Teichmüller codes

Let q = 2 r and R = GR(4, r) the Galois ring of characteristic 4 and residue field

F 2 r . The order of R is q 2 . For integers k ≥ 2 and s ∈    {0, 2, 4, . . . , (k -1)r} if k odd;
{r, r + 2, r + 4, . . . , (k -1)r} if k even, the generalized Teichmüller codes T q,k,s are constructed in [START_REF] Kiermaier | Geometrische Konstruktionen linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener Minimaldistanz[END_REF]Sec. 3.1], [START_REF] Kiermaier | Generalized Teichmüller codes[END_REF], generalizing the Teichmüller codes in [START_REF] Kiermaier | New ring-linear codes from dualization in projective Hjelmslev geometries[END_REF]. They are regular projective R-linear codes of length n = 2 s • q k -1 q-1 and with three non-zero homogeneous weights, scaled by the factor 1/q r-2 , 2 w = 2 s q k -2 s/2 q (k-1)/2 , 2 s q k , 2 s q k + 2 s/2 q (k-1)/2 of frequencies A = 1 2 (q k -1)(q k + 2 s/2 q (k+1)/2 ), q k -1, 1 2 (q k -1)(q k -2 s/2 q (k+1)/2 ) .

From the length we get 2 s q k = n(q -1). Thus, the sum of the weights is S = 3•2 s q k q r-2 = 3(n(q -1)+2 s )q r-2 = 3 q (b+n(q -1)q r-1 ) for b = 2 s q r-1 . By Theorem 12 we see that the coset graph with b loops on each vertices is a 3-SWRG.

We remark that in [START_REF] Kiermaier | New ring-linear codes from dualization in projective Hjelmslev geometries[END_REF] and [START_REF] Kiermaier | Geometrische Konstruktionen linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener Minimaldistanz[END_REF], more projective three-weight-codes over Galois rings are found, among them the dualized extended Kerdock codes K * k+1 and the dualized generalized Teichmüller codes T * q,k,s . However, they are either not regular or the sum of the weights does not satisfy the divisibility conditions needed in the context of this paper. In a similar manner as what we did over Z 4 , we classify 3-weight codes with dual weight d ⊥ ≥ 3 over the other proper chain ring of order 4, which is the ring R = F 2 + uF 2 of dual numbers over F 2 . The Gray image of such a code is always a linear binary code, and a linear binary code is the Gray image of an R-linear code if and only if it has a fixed-point free automorphism of order 2 [START_REF] Honold | All Reed-Muller codes are linearily representable over the ring of dual numbers over Z 2[END_REF]. We classify the codes in the same range of parameters as in the Z 4 -case, that is n ≤ 10, S ≥ 3n and S ≡ 0 mod 3. Since the MacWilliams identities for R-linear codes are the same as for Z 4 -linear codes (and up to doubling the length, the same as the MacWillimans identities for the linear Gray image), we can start with the same list of feasable parameters we computed for Z 4 -linear codes.

The feasable parameters, together with the weight distribution and the value S -3n ≥ 0, are listed in Table 4. The column (k 1 , k 2 ) lists the shapes of the realizable codes. An entry "-" indicates that no R-linear code with the given parameters does exist.

A list of suitable generator matrices for the realizable parameters is given in Table 5.

We remark that we found further examples of R-linear three-weight codes with d ⊥ ≥ 3, S -3n ≥ 0 and 3 | S. Their parameters are displayed in Table 6 6

Conclusion and open problems

In this work we have generalized from fields to rings the constructions of SWRGs from [START_REF] Shi | Three weight codes, triple sum sets and strongly walk regular graphs[END_REF]. The rings considered here are Z p m , and more generally finite chain rings. The weight playing the role of the Hamming in this generalization is the homogeneous weight. This weight can defined in general over finite Frobenius rings. It would be of theoretical interest to consider three-weight codes over those rings.

We have classified short length three-weight Z 4 -codes leading to SWRGs. To undertake such a classification for higher lengths and other rings is a challenging open problem, both theoretically and computationally. The polynomial analogues of Z m p , namely the rings F p [x]/(x m ), are worth considering.

Definition 2 .

 2 Let C have × n generator matrix G = [g 1 | . . . |g n ]. The code C is called:

1 p;Proof.

 1 For the first two cases we apply Theorem 2 to conclude the equivalence to Γ b C(Ω) being a 3-SWRG for b = 0 and b = 1, respectively. Combining Corollary 1 with Theorem 5 gives the third case, which contains the first two cases for b = 0 and b = 1, respectively.

Example 1 :

 1 If p = m = 2 and n = 4, then the possible triples of weights in the first case of Theorem 6 are, with simplified notation, and by increasing value of w 2 equal to 129, 138, 237, 147, 246, 345, 156, 255.

Example 4 :

 4 If n = 12, p = m = 2 the only triples (w 1 , w 2 , w 3 ) satisfying the first condition of Theorem 8 for s = 5 are {11 -i, 12, 13 + i}, for i = 0, 1, . . . , 10.

Proposition 1 .

 1 If C is a Z 4 -code with three Lee weights satisfying the hypotheses of Theorem 6, then its binary image C G satisfies the hypotheses of Theorem 6. Proof. As is well-known, the Hamming weights of C G are the Lee weights of C. If n is the length of C then C G is of length 2n. The third condition of Theorem 6 becomes with p = m = 2,

Lemma 4 .

 4 Let C be an [n, (k 1 , k 2 )] 3-weight code with d ⊥ ≥ 3 and weights w 1 , w 2 , and w 3 satisfying w 1 + w 2 + w 3 = 3n. Then, n ≡ 0 (mod 2).

5. 3

 3 Classification of 3-weights codes over F 2 + uF 2 with d ⊥ ≥ 3 and short length

Table 1 :

 1 Attainable parameters for S ≡ 0 (mod 3)

	, 1) → G 2 5,1

Table 2 :

 2 Computationally excluded cases for S = 3n

	n 2k 1 + k 2 w	A	(k 1 , k 2 )
	3 5	(2,4,6)	(15,15,1)	(1,3)
	5 5	(4,6,8)	(16,12,3)	(1,3)
	7 5	(6,8,10) (16,11,4)	(2,1), (1,3)
	7 6	(6,8,10) (42,7,14)	(2,2), (1,4)
	7 7	(4,8,12) (31,95,1)	(3,1), (2,3), (1,5)
	7 8	(4,8,12) (65,187,3) (4,0), (3,2), (2,4), (1,6)
	9 5	(8,10,12) (15,12,4)	(1,3)
	9 6	(8,11,14) (43,16,4)	(3,0), (2,2), (1,4)
	10 7	(8,12,16) (62,64,1)	(2,3), (1,5)
	10 8	(8,12,16) (130,120,5) (3,2), (2,4), (1,6)

Table 3 :

 3 Computationally excluded cases for S > 3n, S ≡ 0 (mod 3)

Note that Equation[START_REF] Delsarte | Weights of linear codes and strongly regular normed spaces[END_REF] has infinitely many solutions for t ∈ {-1, 0, 1}. Also if y is not a power of

there are lots of solutions, e.g. (x, t, y) ∈ {(1, 2, 15),[START_REF] Van Dam | Strongly walk-regular graphs[END_REF][START_REF] Brouwer | Distance-regular graphs[END_REF][START_REF] Honold | Characterization of finite Frobenius rings[END_REF],(14,

[START_REF] Brouwer | Spectra of graphs[END_REF][START_REF] Shi | A note on one weight and two weight projective Z 4 -codes[END_REF],[START_REF] Greferath | Finite-ring combinatorics and MacWilliams' equivalence theorem[END_REF][START_REF] Calderbank | The geometry of two-weight codes[END_REF] 145)}.

The scaling of the homogeneous weight in[START_REF] Kiermaier | Geometrische Konstruktionen linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener Minimaldistanz[END_REF] and[START_REF] Kiermaier | New ring-linear codes from dualization in projective Hjelmslev geometries[END_REF] differs by ours by the factor q r-2 .

* This research is supported by National Natural Science Foundation of China (12071001, 61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Academic fund for outstanding talents in universities (gxbjZD03)

1 0 1 0 X 0 1 1 1 1 0 0 0 X X 6 5 4 6 6 16 8 9 0 (2, 1)

6 6 4 18 6 24 8 21 0 (3, 0)

6 6 4 18 6 24 8 21 0 (2, 2) 3 (2, 1)

10 8 8 130 12 120 16 5 6 (4, 0)