Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs

In this paper, we consider additive circulant graph codes which are self-dual additive \begin{document}$ \mathbb{F}_4 $\end{document} -codes. We classify all additive circulant graph codes of length \begin{document}$ n = 30, 31 $\end{document} and \begin{document}$ 34 \le n \le 40 $\end{document} having the largest minimum weight. We also classify bordered circulant graph codes of lengths up to 40 having the largest minimum weight.


Introduction
Let F 2 = {0, 1} be the finite field of two elements and F 4 = {0, 1, ω, ω 2 } be the finite field of four elements where ω 2 = ω + 1. An additive F 4 -code C of length n is an additive subgroup of F n 4 . An additive (n, 2 k ) F 4 -code C is a code of length n which contains 2 k codewords. A generator matrix of an additive (n, 2 k ) F 4 -code C is a k ×n matrix whose rows are a basis of C. The weight of a vector c is the number of nonzero components of c. The minimum weight of a code C is the smallest weight among all nonzero codewords of C. An additive (n, 2 k ) F 4 -code having minimum weight d is called an additive (n, 2 k , d) F 4 -code.
The trace inner product of two vectors x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ F n 4 is defined by If C is an additive (n, 2 k ) F 4 -code, its dual code C * = {x ∈ F n 4 | x * y = 0 for all y ∈ C} is an additive (n, 2 2n−k ) F 4 -code. A code C is self-dual if C = C * . A self-dual additive F 4 -code of length n is an (n, 2 n ) F 4 -code. For any length n, there is a self-dual additive F 4 -code. Self-dual additive F 4 -codes can be applied to designing DNA codes for use in DNA computing and solving problems of DNA codes which satisfy some constraints [11]. A self-dual additive (n, 2 n , d) F 4 -code C gives a quantum [[n, 0, d]] code (see [3] for a description of quantum codes). These are motivations for our study of self-dual additive F 4 -codes.
Two additive F 4 -codes C 1 and C 2 are called equivalent if there is a map sending the codewords of C 1 onto the codewords of C 2 where the map consists of a permutation of coordinates, followed by multiplication of coordinates by nonzero elements of F 4 , followed by possible conjugation of the coordinates. The conjugation of x ∈ F 4 is defined by x = x 2 . All self-dual additive F 4 -codes of length n were classified by Calderbank, Rains, Shor and Sloane [3] for n ≤ 5. All self-dual additive F 4 -codes of length n were classified by using n × n adjacency matrices of graphs for 1 ≤ n ≤ 12, by Danielsen and Parker [5]. Varbanov [10] constructed some self-dual additive F 4 -codes from adjacency matrices of circulant graphs. All additive circulant graph codes of length 13 ≤ n ≤ 29 and 31 ≤ n ≤ 33 having the largest minimum weight were classified by Varbanov [10]. All bordered circulant graph codes of length n = 2, 3,6,8,9,14,15,18,20,22 having the largest minimum weight were classified by Danielsen and Parker [6].
A graph code is an additive F 4 -code with generator matrix Γ + ωI where Γ is the adjacency matrix of a graph and I is the identity matrix. In this paper, we classify all additive circulant graph codes having the largest minimum weight for length n = 30 and 34 ≤ n ≤ 40, and classify all bordered circulant graph codes having the largest minimum weight for length n = 4, 5, 7, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, . . . , 40. All computer calculations were done using Magma [2].

Self-dual additive F 4 -codes from graphs
A self-dual additive F 4 -code is called Type II if all codewords have even weights, it is called Type I otherwise. It is known that there is a Type II additive F 4 -code of length n if and only if n is even.
A (simple) graph is a pair (V, E) where V = {v 1 , . . . , v n } is a finite set of vertices, and E is a set of edges. Here, an edge is a 2-subset of V . The adjacency matrix of a graph (V, E) is an n × n F 2 -matrix (a ij ) where a ij = a ji = 1 if {v i , v j } ∈ E, and a ij = a ji = 0 otherwise. Let Γ denote the adjacency matrix of a graph. Then Γ is a symmetric matrix with the diagonal elements are all zero.
Any graph code is self-dual [5]. It was shown that for any self-dual additive F 4 -code C, there is a graph code C(Γ) such that C and C(Γ) are equivalent [5]. This means that self-dual additive F 4 -codes can be represented by the adjacency matrices of some graphs. We can restrict our study to self-dual additive F 4 -codes with generator matrices of the form Γ + ωI. In [5], all self-dual additive F 4 -codes of length n were classified by classifying graphs with n vertices for n ≤ 12.
It seems to be hard to give a classification and determine the largest minimum weight for all self-dual additive F 4 -codes of length 13 or more. We consider only a special form of an adjacency matrix of a graph. An n × n matrix of the form is called a circulant matrix. A graph G is called a circulant graph if the adjacency matrix of G is circulant. Circulant graphs have been studied widely (see e.g., [1,4,7]). Varbanov [10] focused on constructing additive F 4 -codes from circulant graphs to restrict the search space. A graph code C(Γ) is called an additive circulant graph code if Γ is circulant. A symmetric matrix of the form (1) has the property that b i = b n−i (i = 1, . . . , n/2 ). Thus, an additive circulant graph code C(Γ) depends on the first n/2 coordinates (b 1 , . . . , b n/2 ). Danielsen and Parker [6] considered graphs with the following n × n adjacency matrices: where Γ are the adjacency matrices of graphs with n − 1 vertices (more generally, Danielsen and Parker [6] considered additive F 4 -codes C(Γ * ) with generator matrices Γ * + ωI where Γ * are the adjacency matrices of directed graphs). Let C(Γ) denote the additive F 4 -code with generator matrix Γ + ωI. We call C(Γ) a bordered circulant graph code. Any bordered circulant graph code is self-dual since it is a graph code. Proposition 1. A bordered circulant graph code of even length is always Type II.
Proof. Consider a bordered circulant graph code C(Γ) of length n where Γ is the adjacency matrix of a graph with n − 1 vertices. Suppose that n is even. Then the first row (b 0 , b 1 , . . . , b n−2 ) of Γ has even weight since Γ is symmetric and the first row satisfies the property b i = b n−1−i (i = 1, . . . , n−2 2 ). Thus, each row of Γ + ωI has even weight. It is shown that a graph code C(Γ ) is Type II if and only if all the vertices of a graph with adjacency matrix Γ have odd degrees, in other words, each row of Γ + ωI has even weight [5]. Therefore, C(Γ) is Type II.
To obtain all inequivalent codes among the constructed additive circulant graph codes, we use the following method, by Calderbank, Rains, Shor and Sloane [3]. We map the additive (n, 2 k ) F 4 -code C to the binary [3n, k] code β(C) by applying the map β : 0 → (000), 1 → (011), ω → (101), ω 2 → (110) to the coordinates of C. Then, two self-dual additive F 4 -codes C and C are equivalent if and only if the two binary codes β(C) and β(C ) are equivalent.

Additive circulant graph codes of lengths up to 40
Let d A max (n) denote the largest integer d such that a circulant graph code C(Γ) of length n which has minimum weight d exists. In this section, we give a classification of additive circulant graph codes of length n having the largest minimum weight d A max (n) for 1 ≤ n ≤ 12, n = 30, 31 and 34 ≤ n ≤ 40. Varbanov [10] determined d A max (n) for n ≤ 33. Grassl and Harada [8] determined d A max (n) for 34 ≤ n ≤ 50. All additive circulant graph codes of length 13 ≤ n ≤ 29 and 31 ≤ n ≤ 33 having the largest minimum weight d A max (n) were classified, up to equivalence, by Varbanov [10].
By exhaustive computer search, we found all distinct Type I and Type II additive circulant graph codes C(Γ) having the largest minimum weight d A max (n) for length 1 ≤ n ≤ 12, n = 30 and 34 ≤ n ≤ 40. Then, by the method described in Section 2, we determined by Magma [2] whether two additive circulant graph codes are equivalent or not. Then we have a classification of the additive circulant graph codes of length n having minimum weight d A max (n) for 1 ≤ n ≤ 12, n = 30, 31 and 34 ≤ n ≤ 40 where d A max (n) is listed in Table 1. Let num A I (n) and num A II (n) denote the numbers of inequivalent Type I and Type II additive circulant graph codes of length n having the minimum weight d A max (n), respectively. To save space, we only list num A I (n) and num A II (n) in Table 1. The fifth and tenth columns of We give an observation of some codes given in Table 1. The five inequivalent Type I additive circulant graph codes with parameters (31, 2 31 , 10) in Table 1 are constructed as the codes C(Γ  Table 2 where A i denotes the number of codewords of weight i. The weight distributions also yield that these codes are inequivalent. The unique additive circulant graph code with parameters (30, 2 30 , 12) in Table 1 is constructed as the code C(Γ 30 ) where Γ 30 is the 30 × 30 circulant matrix with the following first row: We verified by Magma [2] that the code C(Γ 30 ) is equivalent to the extended quadratic residue code Q 30 described in [9]. The weight distribution of Q 30 is given in [9, Table I]. The weight distribution of the unique additive circulant graph code with parameters (36, 2 36 , 11) in Table 1 is given in [8, Table 2].

Bordered circulant graph codes of lengths up to 40
Let d B max (n) denote the largest integer d such that a bordered circulant graph code C(Γ) of length n which has minimum weight d exists. In this section, we give a classification of bordered circulant graph codes of length n having the largest minimum weight d B max (n) for n = 4, 5, 7, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, . . . , 40. As described above, Danielsen and Parker [6] considered additive F 4 -codes constructed from not only graphs but also directed graphs. It follows that all bordered circulant graph codes for length n = 2, 3,6,8,9,14,15,18,20,22 having the largest minimum weight d B max (n) were classified.
By exhaustive computer search, we determined the largest minimum weight d B max (n) of bordered circulant graph codes of length n while we found all distinct bordered circulant graph codes of length n having the largest minimum weight d B max (n) for n = 4, 5, 7, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, . . . , 40. Then, by the method described in Section 2, we classified all the bordered circulant graph codes of length n having minimum weight d B max (n) where d B max (n) is listed in Table 3. Let num B (n) denote the number of inequivalent bordered circulant graph codes of length n having the minimum weight d B max (n). To save space, we only list num B (n) for 2 ≤ n ≤ 40 in Table 3. The fourth and eighth columns of Table 3 provide references for d B max (n) and num B (n). For the bordered circulant graph codes C(Γ) in Table 3 We give an observation of some codes given in Table 3. We verified by Magma [2] that the unique bordered circulant graph code with parameters (18, 2 18 , 8) in Table 3 is equivalent to the extended quadratic residue code S 18 described in [9]. Danielsen and Parker [6] constructed a self-dual additive F 4 -code with parameters (30, 2 30 , 12) by bordering a quadratic residue code. The unique bordered circulant graph codes with parameters (29, 2 29 , 9), (30, 2 30 , 12), (31, 2 31 , 10) and (37, 2 37 , 11) in Table 3 are constructed as the codes C(Γ 28 ), C(Γ 29 ), C(Γ 30 ) and C(Γ 36 ), respectively, where Γ n−1 are the (n − 1) × (n − 1) circulant matrices with the following first rows r n−1 (n = 29, 30, 31, 37): We verified by Magma [2] that the code C(Γ 29 ) is equivalent to Q 30 . We calculated by Magma [2] the weight distribution of each code C(Γ n−1 ) (n = 29, 30, 31, 37). The weight distributions are listed in Table 4.