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On the classification of self-dual [20,10,9] codes over GF(7)

It is shown that the extended quadratic residue code of length 20 over GF(7) is a unique self-dual [20,10,9] code C such that the lattice obtained from C by Construction A is isomorphic to the 20-dimensional unimodular lattice D_{20}^+, up to equivalence. This is done by converting the classification of such self-dual codes to that of skew-Hadamard matrices of order 20.


Introduction
Let GF(p) be the finite field of order p, where p is prime. As described in [16], self-dual codes are an important class of linear codes for both theoretical and practical reasons. For p ≡ 1 (mod 4), a self-dual code of length n over GF(p) exists if and only if n is even, and for p ≡ 3 (mod 4), a self-dual code of length n over GF(p) exists if and only if n ≡ 0 (mod 4). It is a fundamental problem to classify self-dual codes over GF(p) and determine the largest minimum weight among self-dual codes over GF(p) for a fixed length. Much work has been done towards classifying self-dual codes over GF(p) and determining the largest minimum weight among self-dual codes of a given length over GF(p) for p = 2 and 3 (see [16]).
Self-dual codes over GF (7) have been classified for lengths up to 12 (see [9]), and the largest minimum weight d 7 (n) among self-dual codes of length n over GF (7) has been determined for n ≤ 28 (see [7,Table 2]). For example, it is known that d 7 (20) = 9 and the extended quadratic residue code QR 20 of length 20 over GF (7) is a self-dual [20, 10,9] code (see [5]).
There are 12 nonisomorphic 20-dimensional unimodular lattices having minimum norm 2 (see [3, Table 16.7]), and one of them is D + 20 . Let A 7 (C) denote the unimodular lattice obtained from a self-dual code C over GF (7) by Construction A.
In this paper, we convert the classification of self-dual [20, 10,9] codes C over GF (7) such that A 7 (C) is isomorphic to D + 20 to that of skew-Hadamard matrices of order 20. The main aim of this paper is to give the following partial classification of self-dual [20, 10,9] codes over GF (7). Theorem 1. Up to equivalence, the extended quadratic residue code of length 20 over GF (7) is a unique self-dual [20, 10,9] code C over GF (7) such that A 7 (C) is isomorphic to D + 20 . All computer calculations in this paper were done with the help of Magma [1].

Preliminaries
In this section, we give definitions and notions on self-dual codes, unimodular lattices and skew-Hadamard matrices. Some basic facts on these subjects are also provided.
2.1. Self-dual codes. An [n, k] code C over GF(p) is a k-dimensional subspace of GF(p) n . The value n is called the length of C. The weight wt(x) of a vector x ∈ GF(p) n is the number of non-zero components of x. A vector of C is called a codeword of C. The minimum non-zero weight of all codewords in C is called the minimum weight of C and an [n, k] code with minimum weight d is called an [n, k, d] code. The weight enumerator W (C) of C is given by W (C) = n i=0 A i y i , where A i is the number of codewords of weight i in C. The dual code C ⊥ of C is defined as 2.2. Unimodular lattices. An n-dimensional (Euclidean) lattice is a discrete subgroup of rank n in R n . A lattice L is unimodular if L = L * , where the dual lattice L * is defined as under the standard inner product (x, y). The norm x 2 of a vector x ∈ R n is (x, x). The minimum norm of L is the smallest norm among all nonzero vectors of L. Two lattices L and L ′ are isomorphic, denoted L ∼ = L ′ , if there exists an orthogonal matrix A with L ′ = {xA | x ∈ L}.
Let C be a code of length n over GF(p) and let ε 1 , . . . , ε n be an orthogonal basis of R n satisfying (ε i , ε j ) = pδ i,j , where δ i,j is the Kronecker delta. Then we define the lattice A p (C) obtained from C by Construction A as It is known that Conversely, if a unimodular lattice L contains a p-frame, then there is a self-dual code C over GF(p) with A p (C) ∼ = L (see [8]). Let C be a self-dual [20, 10, d] code over GF (7) with d ∈ {8, 9}. Then it is easy to see that A 7 (C) has minimum norm 2. It is known that there are 12 nonisomorphic 20-dimensional unimodular lattices having minimum norm 2 (see [3, Table 16.7]), and one of them is D + 20 . The lattice D + 20 is defined from the root lattice D 20 as follows: where e i = (δ 1,i , . . . , δ 20,i ) (1 ≤ i ≤ 20) and 1 denotes the all-one vector. Note that D 20 is the even sublattice of D + 20 , that is, the sublattice consisting of vectors of even norm in D + 20 .

Skew-Hadamard matrices.
A Hadamard matrix of order n is an n×n (1, −1)-matrix H such that HH ⊤ = nI, where I is the identity matrix and H ⊤ denotes the transposed matrix of H. It is well known that the order n is necessarily 1, 2, or a multiple of 4. Two Hadamard matrices H and K are said to be equivalent if there are (1, −1, 0)monomial matrices P and Q with K = P HQ. All Hadamard matrices of orders up to 32 have been classified (see [10,Chap. 7] for orders up to 28 and [11] for order 32, see also [18]). A Hadamard matrix H of order n is called a skew-Hadamard matrix if H +H ⊤ = 2I. Skew-Hadamard matrices are a class of Hadamard matrices, which has been widely studied (see e.g., [4], [13]). The numbers of inequivalent skew-Hadamard matrices of orders 4, 8, 12, 16, 20, 24 are 1, 1, 1, 3, 2, 11, respectively [13]. We denote by S 1 the Paley Hadamard matrix of order 20, which is a skew-Hadamard matrix. The other skew-Hadamard matrix of order 20 can be found in [12] and we denote the matrix by S 2 . Moreover, we have verified with the help of Magma that S 2 is equivalent to had.20.toncheviv in [18].
The following lemma can be proved in the same manner as [15,Lemma 3].
Lemma 2. Let F be a square matrix all of whose entries are integers. If F F ⊤ = kI and p is a prime divisor of k such that p 2 ∤ k, then F generates a self-dual code over GF(p).
Hence, the code over GF(7) generated by the row vectors of H + 2I is self-dual, where H is a skew-Hadamard matrix of order 20.

Proof of Theorem 1
In this section, we give a proof of Theorem 1, which is the main result of this paper.
Since f h,i are odd integers, we see that there exists a unique h i such that f h i ,i = ±3. Since F F ⊤ = 28I, the mapping i → h i is a bijection from {1, . . . , 20} to itself. Now we may assume without loss of generality Then all the entries of H are ±1, and the diagonal entries are 1.
We claim H +H ⊤ = 2I. To prove this, we need to show f h,i +f i,h = 0 Similarly, These contradict Lemma 3, and complete the proof of the claim. Since Thus the i-th row of F = H + 2I belongs to C. The fact that F generates the self-dual code C follows from Lemma 2.
We say that skew-Hadamard matrices H and H ′ of order n are skew-Hadamard equivalent if there exists a (1, −1, 0)-monomial matrix P with P HP ⊤ = H ′ . Let H and H ′ be skew-Hadamard matrices of order 20. Let C(H) denote the code over GF(7) generated by the row vectors of H + 2I. If H and H ′ are skew-Hadamard equivalent, then C(H) and C(H ′ ) are equivalent. By Proposition 4, we can convert the classification of self-dual [20, 10,9] codes C over GF(7) with A 7 (C) ∼ = D + 20 to that of skew-Hadamard matrices of order 20, up to skew-Hadamard equivalence. The existence of a skew-Hadamard matrix of order n is equivalent to the existence of a doubly regular tournament of order n − 1 [17]. It is known that there are two doubly regular tournaments of order 19, up to isomorphism (see [14]). This implies that there are two skew-Hadamard matrices of order 20, up to skew-Hadamard equivalence. Indeed, let H be a skew-Hadamard matrix of order 20 and let D be the diagonal matrix whose diagonal entries are the first row of H. Then Here the 19 × 19 (1, 0)-matrix (M + J)/2 − I is the adjacency matrix of a doubly regular tournament of order 19, where J is the 19 × 19 all-one matrix. Hence, isomorphic doubly regular tournaments of order 19 give skew-Hadamard matrices of order 20, which are skew-Hadamard equivalent. The matrices S 1 and S 2 give the two skew-Hadamard matrices of order 20, up to skew-Hadamard equivalence. We have verified with the help of Magma that the two self-dual codes C(S 1 ) and C(S 2 ) have the following weight enumerators: W (C(S 1 )) =1 + 6840y 9 + 47880y 10 + 200640y 11 + 957600y 12 + 3625200y 13 + 10766160y 14 + 25701984y 15 + 48495600y 16 + 68276880y 17 + 68299680y 18 + 43155840y 19 + 12940944y 20 , W (C(S 2 )) =1 + 1080y 8 + 5040y 9 + 40320y 10 + 215760y 11 + 977040y 12 + 3571200y 13 + 10751040y 14 + 25814304y 15 + 48431880y 16 + 68208840y 17 + 68403000y 18 + 43106160y 19 + 12949584y 20 , respectively. In particular, C(S 1 ) is a [20, 10,9] code, while C(S 2 ) has minimum weight 8. By Proposition 4, C(S 1 ) is a unique selfdual [20, 10,9] code C over GF(7) with A 7 (C) ∼ = D + 20 . In addition, we have verified with the help of Magma that A 7 (QR 20 ) ∼ = D + 20 . This completes the proof of Theorem 1.
Remark 5. The above argument yields that C(S 1 ) is equivalent to QR 20 . A general case including this fact was described in [2, p. 1041] without proof.
4. Some other constructions of self-dual [20, 10,9] codes Finally, in this section, we investigate some other constructions of self-dual [20, 10,9] codes over GF (7). Note that D + 20 is the unique 20-dimensional unimodular lattice with minimum norm 2 and kissing number 760. For a given self-dual [20, 10,9] code C over GF (7), one can determine whether A 7 (C) is isomorphic to D + 20 or not, by computing the kissing number of A 7 (C) with the help of Magma. If A 7 (C) is isomorphic to D + 20 , then by Theorem 1, we have that C is equivalent to QR 20 .
A [20, 10] code over GF (7) with the following generator matrix is called a four-circulant (resp. four-negacirculant) code. By exhaustive search, we have verified that A 7 (C) ∼ = D + 20 for all four-circulant self-dual [20, 10,9] codes C. Also, we have verified that A 7 (C) ∼ = D + 20 for all four-negacirculant self-dual [20, 10,9] codes C. • Let C be a self-dual code of length 20 over GF (7). Let x be a vector with x · x = 0. Then C(x) = C ∩ x ⊥ , x is a selfdual code over GF (7). By exhaustive search, we have verified that A 7 (QR 20 (x)) ∼ = D + 20 for all vectors x in a set of complete representatives of GF(7) 20 /QR 20 with x · x = 0. Moreover, our extensive search failed to discover a self-dual [20, 10,9] code C over GF(7) with A 7 (C) ∼ = D + 20 . We are lead to conjecture that QR 20 is a unique self-dual [20, 10,9] code over GF (7).