SELF-ORTHOGONAL CODES FROM THE STRONGLY REGULAR GRAPHS ON UP TO 40 VERTICES

. This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group G which acts with orbits of length w , where w divides | G | . We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters (36 , 15 , 6 , 6), (36 , 14 , 4 , 6), (35 , 16 , 6 , 8) and their complements, and from the graphs with parameters (40 , 12 , 2 , 4) and their complements. That completes the classi- ﬁcation of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of 2-(27 , 9 , 4) designs obtained as residual designs of the symmetric (40 , 13 , 4) designs (complementary designs of the symmetric (40 , 27 , 18) designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.


Introduction
An active area of research in coding theory is the construction and classification of self-orthogonal codes of small lengths and dimensions, see for example [5,6] or [15] and its references. The use of orbit matrices of block designs for constructing self-orthogonal codes was initiated by Harada and Tonchev in [15]. Recently, in [9] studies were carried to further examine an interplay between codes and orbit matrices of block designs. It was established in particular that under certain conditions the codes spanned by the non-fixed parts of the orbits matrices of symmetric designs are self-orthogonal. In this paper we introduce the use of orbit matrices of strongly regular graphs for the construction of self-orthogonal codes. The purpose of this paper is two-fold: First, we present a method of constructing self-orthogonal codes from orbit matrices of strongly regular graphs with respect to an automorphism group G which acts with orbits of length w, where w divides |G|. This method explores the notion of an orbit matrix of a strongly regular graph proposed by Behbahani and Lam in [2]. Secondly, using the introduced method we give a classification of binary and ternary self-orthogonal codes obtained from orbit matrices of the strongly regular (40, 12,2,4), (36,15,6,6), (36,14,4,6), (35, 16,6,8) graphs and their complements. Furthermore, we examine the ternary codes defined by the incidence matrices of the symmetric (40, 27,18) designs corresponding to the strongly regular (40, 27,18,18) graphs, and those of the residual designs of their complementary designs. We also examine the ternary hulls of the residual designs. Moreover, we study ternary codes spanned by adjacency matrices of strongly regular graphs with parameters (36, 15,6,6), (36,14,4,6) and (35,18,9,9). Together with the results of W. H. Haemers, R. Peeters and J. M. van Rijckevorsel [13], that completes the classification of self-orthogonal codes spanned by the adjacency matrices and orbit matrices of the strongly regular graphs on up to 40 vertices. Construction of codes from adjacency matrices of graphs can be considered a special case of the construction of codes from orbit matrices, because the adjacency matrix of a graph Γ is the orbit matrix of Γ with respect to the trivial group.

Background and terminology
We assume that the reader is familiar with some basic notions and elementary facts from design and coding theories, see for example [1] or [3]. An incidence structure D = (P, B, I), with point set P, block set B and incidence I ⊆ P × B, is a t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and every t distinct points are together incident with precisely λ blocks. The complement of D is the structureD = (P, B,Ĩ), whereĨ = P × B − I. The design is symmetric if it has the same number of points and blocks. In a 2-(v, k, λ) design every point is incident with exactly r = λ(v − 1) k − 1 blocks, and r is called the replication number of a design. The number n = r − λ is called the order of a 2-(v, k, λ) design. If a design is symmetric, then r = k. Given a symmetric 2-(v, k, λ) design D, a residual design of D with respect to a block is the design obtained by deleting a block of D and retaining those points not incident with the block. A residual design at any block of D is a 2-(v − k, k − λ, λ) design. A derived design of D with respect to a block is the design obtained by deleting a block and retaining those points incident with the block. A derived design of D with respect to a block is a 2-(k, λ, λ − 1) design. An automorphism of a design D is a permutation on P which sends blocks to blocks. The set of all automorphisms of D forms its full automorphism group denoted by Aut(D).
The code C F of the design D over the finite field F is the vector space spanned by the incidence vectors of the blocks over F . If the point set of D is denoted by P and the block set by B, and if Q is any subset of P, then we will denote the incidence vector of Q by v Q . Thus C F = v B | B ∈ B , and is a subspace of F P , the full vector space of functions from P to F . All our codes will be linear codes, i.e. subspaces of the ambient vector space. If a code C over a field of order q is of length n, dimension k, and minimum weight d = d(C), then we write [n, k, d] q to show this information. An [n, k] linear code C is said to be a best known linear [n, k] code if C has the highest minimum weight among all known [n, k] linear codes. An [n, k] linear code C is said to be an optimal linear [n, k] code if the minimum weight of C achieves the theoretical upper bound on the minimum weight of [n, k] linear codes, and near-optimal if its minimum distance is at most 1 less than the largest possible value. The weight enumerator of C is defined as The dual code C ⊥ is the orthogonal complement under the standard inner product (· , ·), i.e. C ⊥ = {v ∈ F n |(v, c) = 0 for all c ∈ C}. The hull of a design D with code C over the field F is the code obtained by taking the intersection of C and its dual. A linear [n, k] code is called projective if no two columns of a generator matrix G are linearly dependent, i.e., if the columns of G are pairwise different points in a projective (k − 1)-dimensional space. A two-weight code is a code which has exactly two non-zero weights, say w 1 and w 2 . The dual of a two-weight code belongs to the important family of uniformly packed codes. A code C is self-orthogonal if C ⊆ C ⊥ and self-dual if equality is attained. The all-one vector will be denoted by 1, and is the constant vector whose coordinate entries consist entirely of 1's. An automorphism of a code is any permutation of the coordinate positions that maps codewords to codewords. The set of all automorphisms of a code C forms the automorphism group of C denoted by Aut(C). Two codes are equivalent if one of the codes can be obtained from the other by permuting the coordinates and permuting the symbols within one or more coordinate positions.
Terminology for graphs is standard: the graphs, Γ = (V, E) with vertex set V and edge set E, are undirected and the valency of a vertex is the number of edges incident with the vertex. A graph is regular if all the vertices have the same valency; a regular graph is strongly regular of type (v, k, λ, µ) if it has v vertices, valency k, and if any two adjacent vertices are together adjacent to λ vertices, while any two non-adjacent vertices are together adjacent to µ vertices. A strongly regular graph of type (v, k, λ, µ) is usually denoted by srg(v, k, λ, µ).

Self-orthogonal codes from orbit matrices of strongly regular graphs
Orbit matrices of block designs have been frequently used for construction of block designs, see e.g. [17] and [10]. Recently M. Behbahani and C. Lam [2] introduced the concept of orbit matrices of strongly regular graphs. While Behbahani and Lam were interested in orbit matrices of strongly regular graphs admitting an automorphism of prime order, in this paper we give a general definition of an orbit matrix of a strongly regular graph. Further, we use orbit matrices of strongly regular graphs to construct self-orthogonal codes.
Let Γ be a srg(v, k, λ, µ) and A be its adjacency matrix. Suppose an automorphism group G of Γ partitions the set of vertices V into t orbits O 1 , . . . , O t , with sizes n 1 , . . . , n t , respectively. The orbits divide A into submatrices Since the adjacency matrix is symmetric, R = C T . The matrix R is the row orbit matrix of the graph Γ with respect to G, and the matrix C is the column orbit matrix of the graph Γ with respect to G.
The following theorem gives a method for constructing self-orthogonal codes from row orbit matrices of a strongly regular graph Γ with an automorphism group G which acts on the set of vertices of Γ with all orbits of the same length. Note that such an action of an automorphism group G implies that the row orbit matrix of Γ with respect to G is equal to the column orbit matrix. Theorem 1. Let Γ be a srg(v, k, λ, µ) with an automorphism group G which acts on the set of vertices of Γ with v w orbits of length w. Let R be the row orbit matrix of the graph Γ with respect to G. If q is a prime dividing k, λ and µ, then the matrix R generates a self-orthogonal code of length v w over F q . Proof. Let u be a vertex of Γ. We will denote by u the neighbourhood of u, i.e. the set of all vertices adjacent to u. Let x be an element of the orbit O i , and y s be a representative of an orbit O s , for where α is the number of vertices from the orbit O j adjacent to the vertex x.
where β is the number of vertices from O j adjacent to x.
Since r ij n i = c ij n j and all orbits have length w, it follows that r ij = c ij and therefore t s=1 If q divides k, λ and µ, the row span of R over F q is a self-orthogonal code of length v w . Let us assume that a group G of prime order acts as an automorphism group on a strongly regular graph with parameters (v, k, λ, µ). In order to enable a construction of strongly regular graphs with presumed automorphism group, each matrix with the properties of an orbit matrix is called an orbit matrix for parameters (v, k, λ, µ) and a group G (see [2]). Hence, Behbahani and Lam considered as orbit matrices all matrices R = [r ij ] and R T = C = [c ij ] that satisfy t s=1 c is r sj n s = δ ij (k − µ)n j + µn i n j + (λ − µ)c ij n j .
Based on the above given properties, we propose the following definition of orbit matrices of a strongly regular graph for given parameters and orbit lengths distribution.
is called a row orbit matrix for a strongly regular graph with parameters (v, k, λ, µ) and orbit lengths distribution (n 1 , . . . , n t ). A (t × t)-matrix C = [c ij ] with entries satisfying conditions is called a column orbit matrix for a strongly regular graph with parameters (v, k, λ, µ) and orbit lengths distribution (n 1 , . . . , n t ).
Therefore, we can apply Theorem 1 to all orbit matrices in the sense of Definition 1, regardless of whether or not these are induced by an action of an automorphism group on a strongly regular graph. 4. Codes from orbit matrices of strongly regular graphs with parameters (40, 12, 2, 4) and (40, 27,18,18) Applying Theorem 1 we construct binary self-orthogonal codes from orbit matrices for Z 4 or Z 2 acting on srg(40, 12, 2, 4) with all orbits of length four or two, respectively, and ternary self-orthogonal codes from orbit matrices for Z 4 or Z 2 acting on srg(40, 27,18,18) with all orbits of length four or two. Since the considerations are for the cases where all orbits are of the same length the row orbit matrices coincide with the column orbit matrices, and in such cases we call them just orbit matrices.

4.1.
Orbit matrices for Z 4 . Below we give the matrices C 1 , . . . , C 39 , which are the all orbit matrices for Z 4 acting with ten orbits of length four on strongly regular graphs with parameters (40, 12, 2, 4). These are orbit matrices in the sense of Definition 1. Among the orbit matrices C 1 , . . . , C 39 only five are induced by an action of an automorphism group on some of the strongly regular (40, 12, 2, 4) graphs constructed by Spence in [26]. In particular the orbit matrix C 16 can be obtained from the 25 th graph; the orbit matrices C 13 , C 26 and C 31 can be obtained from the 26 th graph, and the orbit matrix C 36 is related to the 27 th graph.
The orbit matrices for Z 4 acting with ten orbits of length four on strongly regular graphs with parameters (40, 27, 18, 18) will be denoted by D i , where i = 1, . . . , 39. Let I n and J n be the identity matrix and the all one matrix of order n, respectively. It is obvious that D i = 4J 10 − C i − I 10 , for i = 1, . . . , 39. Weight distributions and orders of automorphism groups of self-orthogonal codes spanned by the matrices   Table 6] and also [7]. The codes with parameters [10,4,4] 2 are also optimal. The reader will have noticed that the [10, 4, 4] 2 codes are of two types: they are respectively optimal two-weight and three-weight codes. To our knowledge, these optimal codes are new, i.e. we did not find an evidence that two-weight and threeweight [10, 4, 4] 2 codes have been constructed so far.
The orbit matrix for Z 2 acting fixed-point-freely on a srg(40, 27, 18, 18) that corresponds to the orbit matrix M i , i = 1, . . . , 5, is N i = 2J 20 − M i − I 20 . The weight distributions and the orders of the automorphism groups of the self-orthogonal codes spanned by the matrices M i , i = 1, . . . , 5, and the corresponding orbit matrices for the complementary graphs, are given below.
Proposition 2. The binary (resp. ternary) self-orthogonal codes of length 20 obtained from the orbit matrices of the strongly regular (40, 12, 2, 4) graphs (resp. strongly regular (40, 27,18,18) graphs) with Z 2 acting fixed-point freely are divided into five (resp. four) equivalence classes. Two of these codes are optimal.   The adjacency matrix of a strongly regular graph Γ can be viewed as the orbit matrix of Γ with respect to the trivial group, and so Theorem 1 can be applied to the construction of self-orthogonal codes from adjacency matrices of strongly regular graphs. In the sequel we examine the ternary codes spanned by the adjacency matrices of the strongly regular (40, 27,18,18) graphs. The latter are incidence matrices of the corresponding 2-(40, 27, 18) designs.
In [26], Spence proved that there are up to isomorphism exactly 28 strongly regular graphs with parameters (40, 12,2,4). That also solves the enumeration of strongly regular (40, 27,18,18) graphs, as their complementary graphs. The binary codes spanned by the adjacency matrices of strongly regular graphs, and in particular those of the strongly regular (40, 12, 2, 4) graphs have been studied in [13]. That paper showed that the codes of the (40, 12, 2, 4) strongly regular graphs are non-isomorphic. In this section we examine the ternary codes from the strongly regular (40, 27, 18, 18) graphs. A strongly regular (40, 27,18,18) graph is a regular graph on 40 vertices of degree 27 such that each pair of distinct vertices has 18 common neighbours. Since λ = µ = 18, we may associate with every strongly regular (40, 27, 18, 18) graph a (40, 27, 18) design in such a way that the 40 vertices serve as both point and blocks of the design and adjacency in the graph is incidence in the design. Interchanging points and blocks that correspond to the same vertex gives rise to a polarity in the design for which no point is absolute. Conversely, every 2-(40, 27, 18) design with a polarity without absolute points corresponds to a strongly regular (40, 27,18,18) graph. In sections 5 and 6 we will present our results on the ternary codes of the 2-(40, 27, 18) designs, and those of the residual designs of their complementary designs.
Let Γ i , 1 ≤ i ≤ 28, denote the 28 strongly regular (40, 27,18,18) graphs examined in this section. The ordering of the codes follows that for the strongly regular (40, 12, 2, 4) graphs as given by Spence in [26]. The adjacency matrix of Γ i is also the incidence matrix of a symmetric (40, 27, 18) design (with a polarity having no absolute points). We will use both views interchangeably throughout this section. Note that given a design and any prime p, the p-ary code of the design is the code over F p generated by the rows of the incidence matrix (which are the characteristic functions of the blocks). If A is an incidence matrix of a 2-(v, k, λ) design, then it is well known that its code over F p is interesting only when p divides r − λ, the order of the design (see [28,Theorem 1.86]). Since the order of a symmetric (40, 27, 18) design is 9, only the ternary codes of such designs will be of interest for characterization. Thus, taking the ternary row span of the adjacency matrix of Γ i we construct the codes which we examine in the sequel, and denote these C Γi . However, when the context is clear we shall omit the subscript i.
Notice that there is only one simply primitive (primitive but not 2-transitive) group acting on these designs, namely the automorphism group of Γ 6 . This group is isomorphic to the projective symplectic group P ΓS 4 (3) = S 4 (3):Z 2 . Note also that there is only one 2-transitive group, being the full automorphism group of the design Γ 26 (isomorphic to P ΓL 4 (3)). The attentive reader would have noticed the minimality of the 3-rank of the design Γ 26 . The 3-rank 10 of Γ 26 is minimal amongst the designs with the same parameters. According to the famous Hamada's conjecture [14] this is a property enjoyed by the geometric design. Moreover, Γ 26 possesses the largest automorphism group and the ternary row span of this design produce the code with the largest automorphism group.
For each design Γ i we used Magma [4] to construct a code C Γi as the ternary span of the rows of the incidence matrix of Γ i . Information about the codes are listed in Table 5 and Table 6. The first column represents the ordering of the code corresponding to a 2-(40, 27, 18) design, the second column represents the dimension of the code, and from the third to the last column (the top of the table) we list the number of codewords of a given weight in C Γi .
In the following we examine some properties of the codes C Γi : Proposition 3. Let C Γi denote a code obtained from the ternary row span of an adjacency matrix of Γ i , where Γ i is any of the 28 strongly regular graphs with parameters (40, 27,18,18). Then the following holds: (i) C Γi is a self-orthogonal code. Moreover, 1 ∈ C Γi ∩ C ⊥ Γi ; (ii) the minimum weight of C Γi is 6, 9, 12 or 18; (iii) the minimum weight of C Γi ⊥ is 4 unless Γ = Γ 1 or Γ = Γ 27 in which case the minimum weight is 6; Proof. For i = j, consider B i and B j two distinct blocks in a design corresponding to Γ i . Since k = 27 ≡ 0 (mod 3) and |B i ∩ B j | = 18 ≡ 0 (mod 3), (where i, j ∈ {1, . . . , b}, and b and k are respectively the number of blocks and the block size), the adjacency matrix of Γ i spans a self-orthogonal code of length 40. Moreover, since the block size of Γ i is divisible by 3 we have that 1 ∈ C Γi ⊥ . That 1 ∈ C Γi follows since the sum (modulo 3) of all rows of a generator matrix G of C Γi is the all-ones vector. The assertions (ii) and (iii) were both attained computationally. Finally, in each case and with exception of i = 18, 19 using Magma we determined that Aut(Γ i ) ∼ = Aut(C Γi ). For i = 18 or i = 19 the reader will notice from Table 5 that Aut(Γ i ) ⊂ Aut(C Γi ). i Table 6. Weight distribution of the ternary codes of the strongly regular (40, 27,18,18)  Proof. SinceΓ i is the complement of Γ i , the difference of any two codewords in CΓ i is in C Γi . As these differences span a subcode of co-dimension 1 in CΓ i , this subcode must be C Γi . Also, since 1 can be written, in many ways, as the sum of a codeword in CΓ i and a codeword in C Γi , it is also in CΓ i . Now, if α ∈ Aut(C Γi ), then since α(1) = 1 and CΓ i = C Γi , 1 , it follows that α ∈ Aut(CΓ i ). Hence, Aut(C Γi ) ⊆ Aut(CΓ i ). Moreover, calculations with Magma show that |Aut(C Γi )| = |Aut(CΓ i )| for all cases except when i = 18, 19, and the rest follows. Finally, applying the Rudolph's decoding algorithm [24] forΓ i , we have that r+λ−1 Thus, the rows of the incidence matrix ofΓ i can be used as orthogonal parity checks that allow majority decoding up to its full error-correcting capacity for CΓ i ⊥ .
In the following we make some observations related with the codes obtained, and in particular to C Γ6 = [40, 14, 12] 3 and C Γ26 = [40, 10, 18] 3 and their links with combinatorial designs, graphs and groups.
1. The 540 codewords of weight 12 in C Γ6 remain intact under the action of U 4 (2) · Z 2 . The support of a codeword of weight 12 produces a 1-(45, 12, 81) design with 270 blocks, whose code is a [40, 39, 2] 3 . The 80 codewords of weight 4 in C Γ6 ⊥ represent the isotropic lines and their scalar multiples. Under the action of Aut(C Γ6 ⊥ ) ∼ = P ΓS 4 (3) these codewords split into two orbits each of size 40. The stabilizer of an isotropic line or a scalar multiple of it, is a maximal subgroup of P ΓS 4 (3) isomorphic to E 27 :(S 4 × Z 2 ). Furthermore, the support of an isotropic line or its scalar multiple (codewords of weight 4) give rise to the unique irreducible 25-dimensional F 3 -module, hence code K = [40, 25, 4] 3 invariant under P ΓS 4 (3). The hull of K is precisely C Γ6 . Moreover, C Γ6 is an irreducible F 3 -module invariant under the automorphism group. In fact C Γ6 ⊥ = K ⊕ 1 = K, 1 . The irreducibility of C Γ6 and K follow since the 3-modular character table of the group S 4 (3) is completely known (see [18]). It also follow from this that the irreducible 14 and 25-dimensional F 3 -representations are unique. We illustrate a continuation an argument to show the irreducibility of C Γ6 . The proof for the irreducibility of K follows similarly. Since Aut(C Γ6 ) contains S 4 (3), by using the weight enumerator (see Table 5) one can deduce that K, under the action of S 4 (3), does not contain an invariant subspace of dimension 1. Moreover, if C Γ6 were reducible, it would have to contain an invariant subspace E of dimension m where 2 ≤ m ≤ 13. However, calculations with Meat-Axe within Magma [4] show that the module 14 · (1 ⊕ 1 ⊕ 1) (with no trivial submodules) occurs naturally as a submodule of S 4 (3) acting on the cosets of E 27 :S 4 . Hence C Γ6 is the 14-dimensional F 3 -module on which S 4 (3) acts irreducibly. 2. Notice that the code C Γ26 = [40, 10, 18] 3 has the smallest dimension, but the largest minimum weight and automorphism group. The words of weight 18 in C Γ26 can be described geometrically, i.e., they are the differences of the incidence vectors of two planes of order 3 in P G 3 (F 3 ). These planes meet in a line, i.e., four points, so the weight of the difference of incidence vectors is 18 (see [22] and [8]).

The dual code C Γ26
⊥ is a [40, 30, 4] 3 with 260 codewords of minimum weight. This is in fact the code of the 2-(40, 4, 1) design of points and lines in the projective geometry P G 3 (F 3 ), so the automorphism group of the design is P ΓL 4 (F 3 ) ∼ = L 4 (3):Z 2 by the fundamental theorem of projective geometry. The 260 words of weight 4 are the incidence vectors of the lines, and their scalar multiples. The code C Γ26 ⊥ is a projective generalized Reed-Muller code (see [1,Chapter 5]). The code C Γ26 is an optimal code, and its dual code C Γ26 ⊥ has minimum distance only 1 less than the optimal, thus near-optimal (see [12]). Moreover the 2-(40, 4, 1) design is quasi-symmetric with 130 blocks, and with blocks intersecting in 0 or 1 point. It is well-known that quasi-symmetric design give rise to strongly regular graphs (see [25] [40, 29, 6] 3 is an optimal code. Moreover, the designΓ 26 is isomorphic to the design constructed in [11] and it can also be obtained from the point graph of a generalized quadrangle. 5. The words of weight 13 in CΓ 26 also have a geometric description: they are the incidence vectors of the blocks ofΓ 26  6. The ternary codes of the residual 2- (27,9,4) designs According to [20, 8 2-(27, 9, 4) designs. This number is not feasible for any classification purposes. Thus, in this section we restrict our attention to the class of 2- (27,9,4) designs that are obtained as residual designs from complementary designs of 28 non-isomorphic 2 − (40, 27, 18) designs from previous section. Using Magma we have determined that there are 258 such designs which are mutually non-isomorphic. In Table 7 we present the information about orders of the full automorphism groups of these designs.
The incidence matrices of the 258 2- (27,9,4) designs span up to equivalence 249 ternary codes of length 27. A continuation, (see Table 8), we list the number of codes with given parameters and the orders of their respective automorphism groups.
The results of our computations for the ternary codes of the 2- (27,9,4) designs can be summarized in the following: Proposition 5. Let C denote the ternary code of a 2- (27,9,4) design D as defined above. Then the following holds:    [27,10,9] 303264 1 Table 8. Proof. That the vector 1 is in C ⊥ and in C follow since the block size of D is divisible by 3, and moreover r = 13 ≡ 1 (mod 3). Since s ⊥ ≥ r+λ λ > 4 a counting argument can be used to derive a contradiction if s ⊥ = 5, thus showing that the minimum weight is as asserted. The remaining assertions follow from computations with Magma. Table 8 that the code with the smallest dimension, namely C = [27,10,9] 3 achieves the highest minimum weight and the second largest automorphism group. This code is associated with the code of the projective geometry P G 3 (F 3 ), i.e, Γ 26 listed in Table 5. In this case the words of minimum weight are the incidence vectors of the blocks of the design and their scalar multiples. See the remark following Proposition 6 below for a description of the minimum weight codewords, and an interplay between these and the associated geometric objects. Apart from the dual code C ⊥ = [27, 17, 6] 3 of C = [27,10,9] 3 for which a geometric description of the nature of the codewords of minimum weight is given (see item 2 of the remark following Proposition 6), it would be of interest to have a description of the nature of the minimum weight codewords of C ⊥ for the remaining codes listed in Table 8. Proposition 6. Let H denote the ternary hull of a 2- (27,9,4) design D as defined above. Then for H the following properties holds:

Remark 5. Observe from
(a) Up to equivalence, there are precisely 240 ternary hulls of length 27 obtained from D; Proof. Since for H the weights of all its codewords are divisible by 3 it is clear that H is self-orthogonal. That 1 ∈ H follows from Proposition 5. By examining the sum (modulo 3) of all rows of a generator matrix of H ⊥ we conclude that 1 ∈ H ⊥ . Now, from H ⊆ C we deduce that d(C) ≤ l, but part (a) tells us that wt(x) ≡ 0 (mod 3) for x ∈ H, so that l ∈ {3, 6}. Moreover, H ⊆ C ⊥ implies that 6 = d(C ⊥ ) ≤ l. The last assertion follows from computations with Magma.
1. The unique code H = [27,10,9] 3 listed in Table 9 (see also Table 8) is a distance 2 less than optimal. However its dual code H ⊥ = [27, 17, 6] 3 is optimal. Recall from Section 5 that there is only one design on which a 2-transitive group acts. The code H whose parameters are stated above is the code of the affine resolvable 2- (27,9,4) obtained from the design Γ 26 listed in Table 5, see [19] for a discussion on the codes from the affine designs. The words of weight 9 in H have a geometric description, i.e., they are the incidence vectors of the blocks of the corresponding design 2- (27,9,4)  . The block graph of this design is a srg(117, 36, 15,9) isomorphic to the graph given in [8]. For a description of the properties of the ternary codes of this graph and the nature of the codewords of minimum weight, the reader should consult [8, Proposition 5.5]. 3. The dual L ⊥ of L is the code L ⊥ = [27,4,18] 3 . The code L ⊥ is a self-orthogonal and optimal two-weight code with weight distribution Note that a two-weight code is a code which has only 2 non-zero weights w 1 and w 2 and is called projective if its dual distance is at least 3. Moreover, a code with minimum weight d = 3 and covering radius 2 is called uniformly packed if every vector which is not a codeword is a distance 1 or 2 from * The reader is cautioned of a typographical error in the structure description given for this group in [8] a constant number of codewords. It follows that L ⊥ is projective and L is uniformly packed. The existence of a projective two-weight code of dimension k over F q is equivalent to the existence of a set of n points in the projective space P G(k − 1, q) that has two intersection numbers with the hyperplanes in P G (k − 1, q). The existence and uniqueness of a ternary code with parameters of L ⊥ has been known for some time (see [16]). 8. Codes related to the strongly regular (36, 15, 6, 6) graphs There are exactly 32548 strongly regular graphs with parameters (36, 15, 6, 6) (see [21]). These strongly regular graphs are available at E. Spence's web page [27].
[  In Table 10 we give information on ternary codes spanned by the rows of ad- Table 11. Ternary codes of the orbit matrices for Z 2 acting on srg(36, 15,6,6) Up to isomorphism there are 167 orbit matrices for Z 2 acting fixed-point freely on strongly regular graphs with parameters (36, 15, 6, 6). The matrices are available at http://www.math.uniri.hr/~sanjar/structures/. Table 11 gives information on ternary codes spanned by these orbit matrices.
The parameters of self-orthogonal binary linear codes spanned by the orbit matrices of Z 2 acting on strongly regular graphs with parameters (36, 20, 10, 12) (complements of strongly regular (36, 15, 6, 6) graphs) and orders of their automorphism groups are given in Table 12. The codes with parameters [18,4,8] 2 are optimal.
The parameters of self-orthogonal codes spanned by the orbit matrices of the strongly regular graphs with parameters (36, 15, 6, 6) and their complementary graphs, and orders of their automorphism groups, are given in Proposition 7.
Information on self-orthogonal codes spanned by the orbit matrices for Z 3 acting fixed-point freely on the strongly regular graphs with parameters (36, 15, 6, 6) and their complementary graphs are summarized in Proposition 8.
Proposition 8. The binary (resp. ternary) self-orthogonal codes of length 12 obtained from the orbit matrices of the strongly regular (36, 20, 10, 12) graphs (resp. strongly regular (36, 15, 6, 6) graphs) with Z 3 acting fixed-point freely are divided into forty (resp. sixteen) equivalence classes. The codes belonging to eleven equivalence classes are optimal.   Table 22. Ternary codes of the orbit matrices for Z 5 acting on srg(35, 18,9,9) parameters |Aut(C D i )| no. parameters |Aut(C D i )| no.   Table 24. Ternary codes of the orbit matrices for Z 7 acting on srg(35, 18,9,9) Advances in Mathematics of Communications Volume 10, No. 3 (2016), 555-582 binary codes and orders of their automorphism groups are given in Table 23. Information on ternary codes spanned by the orbit matrices for Z 7 acting fixed-point freely on strongly regular graphs with parameters (35, 18,9,9) is given in Table 24.
The results presented in this paper complete the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Orbit matrices used for constructing the codes are available at http://www.math.uniri.hr/~sanjar/structures/.