{'fraction_non_alphanumeric': 0.04744346116027532, 'fraction_numerical': 0.011430678466076696, 'mean_word_length': 3.887087087087087, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
DIRECTED STRONGLY REGULAR GRAPHS AND THEIR CODES

Directed Strongly Regular Graphs (DSRG) were introduced by Duval as a generalization of strongly regular graphs (SRG’s) [4]. As observed in [8] a special case of these are the doubly regular tournaments or equivalently, the skew Hadamard matrices. As the latter already lead to many interesting codes [10] it is natural to consider the more general case of codes constructed from the adjacency matrix of a DSRG. Another motivation is that the codes of SRG’s have been investigated in [2, 7], and the spectra of SRG and DSRG present many similarities, as evidenced by Section 3.3. In this paper, we investigate the dimensions of these codes by using three techniques: orthogonality of codes, determinant factorization, a classical technique in design theory, and last but not least, spectrum of the adjacency matrices in the spirit of [2, 7]. Examples of codes illustrating the bounds, along with their minimum distances are given. While the codes constructed are not, in general, optimal error correcting codes in the distance vs dimension sense,


Introduction
Directed Strongly Regular Graphs (DSRG) were introduced by Duval as a generalization of strongly regular graphs (SRG's) [4]. As observed in [8] a special case of these are the doubly regular tournaments or equivalently, the skew Hadamard matrices. As the latter already lead to many interesting codes [10] it is natural to consider the more general case of codes constructed from the adjacency matrix of a DSRG. Another motivation is that the codes of SRG's have been investigated in [2,7], and the spectra of SRG and DSRG present many similarities, as evidenced by Section 3.3.
In this paper, we investigate the dimensions of these codes by using three techniques: orthogonality of codes, determinant factorization, a classical technique in design theory, and last but not least, spectrum of the adjacency matrices in the spirit of [2,7]. Examples of codes illustrating the bounds, along with their minimum distances are given. While the codes constructed are not, in general, optimal error correcting codes in the distance vs dimension sense, or related to classical algebraic codes, we believe they are of some interest to the combinatorics of DSRG's in the same way that the codes of [1] are relevant to the Combinatorics of designs. In particular, the problem of computing their dimension is of interest in its own right as the difficult Conjecture of Section 3.2 shows.
The material in this paper is organized as follows. Section 2 recalls the needed notions and notations. Then the three subsections of Section 3 contain the three said bounding techniques as well as code parameters. An appendix displays the adjacency matrices of DSRG's corresponding to the codes constructed.

Background
Let A be a v by v matrix with entries in {0, 1} ⊂ Z and having zero diagonal. Thus A is the adjacency matrix of a directed simple graph without loops on v vertices. This graph is a directed strongly regular graph (DSRG) of parameters (v, k, t, λ, µ) if it satisfies the following pair of relations: • AJ = JA = kJ, , where I and J denote the identity and all-one matrices, respectively, of order v. It is easy to check that if A is the adjacency matrix of a DSRG so is J − I − A. These two DSRGs are said to form a complementary pair. When the ring of coefficients of A is a general field F instead of Z, we still denote the matrix by A.
There is a dependency between the parameters: Lemma 2.1. A DSRG having parameters (v, k, t, λ, µ) satisfies the following equation: Proof. Multiplying by sides of the second equation above by J and using AJ = kJ we obtain Proof. This follows since the second equation above is the same as but by using the fact that the determinant is the product of the eigenvalues it is easy to see that

The rank of the adjacency matrix
The following result is folklore in characteristic zero; c.f. [6].
By definition of a DSRG we have that Q(A) = µJ. By AJ = JA = kJ, we see that (A − kI)Q(A) = 0.
Note that the cubic polynomial P (x) (in a general commutative variable x) is an annihilator polynomial of A. Also, P might not be the minimal polynomial of A, since sometimes the minimal polynomial of A can have degree less than three; c.f. [8].
We are concerned with the rank of A over a finite field F q of characteristic p. The least interesting case for coding theory is when A has full rank.
, then A has rank v over F q . Equivalently, det(A) = 0 implies that either k ≡ 0 (mod p), or t ≡ µ (mod p).
Proof. Since P (0) = k(µ − t), only in this case A can have an eigenvalue zero.
Thus, in the rest of the paper, we will only consider the congruence cases of Corollary 3.2, or look at the linear span of a different matrix. We derive orthogonality properties of codes built from A depending on the roots of the polynomial Q.

Orthogonality properties
When X is a general matrix with v columns, we consider the (linear) code C X := X (with generator matrix X) to be the row space of X. Each vector (of length v) in this row space is called a codeword of C X and so v is also the length of C X . The dual code C ⊥ X is defined to be the set of all (row)-vectors orthogonal to C X . It is well-known that for any linear code C, dim(C) + dim(C ⊥ ) = v. Proposition 3.3. If p divides µ and Q(x) ≡ (x − a) 2 (mod p), then the rank of A − aI over F q is at most v/2, and the codes A − aI and A T − aI are orthogonal to each other. In particular, if p divides µ, λ, and t, then the rank of A over F q is at most v/2.
Proof. With the hypotheses we made, Q(x) = (x − a) 2 over F q and Q(A) = 0, so the codes spanned respectively by the rows of A − aI and its transpose are orthogonal to each other and of the same rank. In particular this is true for Using the tables in [8], we can list some examples of parameters where Proposition 3.3 applies. Note that complementary pairs of graphs do not share the same Q(x). These are the cases of q = 2 in Table 1.  Table 2. Binary [n, k, d] Codes from Table 1 codes We describe the codes from Table 1 in Table 2, where A v,k denotes the adjacency matrix of a DSRG of parameters (v, k) in Table 1 and C(B) is the code spanned by the rows of B. A star indicates an optimal code, that is having the largest dimension for given length and distance as per the tables of [5]. In the appendix, we give the explicit A v,k since the table [8] does not give explicit adjacency matrices in many cases.
Now we consider the case where Q has two distinct roots.
, then the codes A − aI and A T − bI are the duals of each other.
The result follows.
Parameters satisfying Proposition 3.4 are the cases of q = 3, 5 in Table 1.

Determinant bounds
Now we give rank estimates.
Proof. The proof follows by the standard invariant factors argument using the Smith normal form of A (see [9, p. 311] for details). If λ = µ, then A 2 = (t − µ)I + µJ, and consequently, by Lemma 2.2, we have det In practice the determinant of A is best computed by taking the product of the eigenvalues with multiplicity. There are exact formulas to compute these as a function of the 5 parameters [6, Thm. 2.2]. Some parameter values where Proposition 3.5 applies are given in Table 3.  We describe the codes from Table 3 in Table 4, where A v,k denotes the adjacency matrix of a DSRG of parameters (v, k) in Table 3, C(B) is the binary or ternary code spanned by the rows of B, and the exact rank of A v,k is computed. A star indicates an optimal code. In the appendix, we give the explicit A v,k since the table [8] does not give explicit adjacency matrices in many cases.
From Table 4 we see that the rank of A over F q is exactly v − r. Hence we conjecture the following.
Conjecture. If p r divides det(A) = 0 exactly, then the rank of A over F p is v − r.

Spectrum of the adjacency matrix
We shall give very general results by adapting the case of SRG's from [3, 13.7.1]. The proof is analogous and omitted. Consider M = A + bJ + cI. If the spectrum of A is k, r, s, then that of M is θ 0 = k + bv + c, θ 1 = r + c, θ 2 = s + c with the same respective multiplicities m 0 , m 1 , m 2 given by 1, f, g.