Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$ are computed, as well as the binary doubly even self-dual codes of length $120$ with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

It is known that for every odd prime p there exists a Hadamard matrix of order 2p + 2 with an automorphism of order p, found by Paley [25], and known in the combinatorial literature as the Paley-Hadamard matrix of type II.If p is an odd prime such that p ≡ −1 (mod 3) then the Paley-Hadamard matrix of type II of order 2p + 2 is a generator matrix of a Pless symmetry code [26], being a ternary self-dual code of length 2p + 2. In the context of ternary codes, we consider the elements 0, 1, −1 of Z as the elements 0, 1, 2 of the finite field F 3 of order 3.By the Assmus-Mattson theorem [4], the Pless symmetry codes of length 2p + 2 for p = 5, 11, 17, 23 and 29 are ternary extremal self-dual codes that support 5-designs [26].The ternary extended quadratic residue codes of length 2p + 2 are ternary extremal selfdual codes that support 5-designs for p = 5, 11, 23 and 29, and these codes have generator matrices, being Hadamard matrices of order 2p + 2 with an automorphism of order p [33].In 2013, Nebe and Villar [24] gave a series of ternary self-dual codes of length 2p + 2 for a prime p ≡ 5 (mod 8).The selfdual codes are also ternary extremal self-dual codes that support 5-designs for p = 5 and 29 [24].Recently, it was shown in [2] that these codes found by Nebe and Villar [24] also have generator matrices, being Hadamard matrices of order 2p + 2 for p ≡ 5 (mod 24).In addition, it was also shown that there are at least two inequivalent Hadamard matrices in a third ternary extremal self-dual code of length 60.This implies that there are at least four inequivalent Hadamard matrices of order 60 formed by codewords of weight 60 in the known three ternary extremal self-dual codes of length 60.In addition, each of these four Hadamard matrices has an automorphism of order 29.This motivates us to classify the Hadamard matrices of order 60 with an automorphism of order 29.
The Paley-Hadamard matrices of type II for p = 3 and p = 5 coincide with the unique, up to equivalence, Hadamard matrices of orders 8 and 12, respectively.The Hadamard matrices of order 2p + 2 with an automorphism of orders p = 7, 11, 13, 17, 19 and 23 have been previously classified up to equivalence.A short summary on these matrices is given in Section 5, along with relevant references.
In this paper, we present the classification of Hadamard matrices of order 2p + 2 with an automorphism of order p for the next two odd primes, p = 29 and 31.As an application, we compute the ternary self-dual codes and selfdual codes over the finite filed F 5 of order 5 spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29, as well as some binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs.In addition, we study the binary self-dual codes defined by Hadamard designs arising from Hadamard matrices of order 64 with an automorphism of order 31.
This paper is organized as follows.In Section 2, we give definitions and some known results about Hadamard matrices, designs and codes used in this paper.We also review a method from [10], [29] and [31] for classifying Hadamard matrices of order 2p + 2 with an automorphism of order p, where p is an odd prime.
In Section 3, we give the classification of Hadamard 2-(59, 29, 14) designs with an automorphism of order 29 having one fixed point, and show that there are exactly 531 non-isomorphic such designs.Using these designs, we classify up to equivalence the Hadamard matrices of order 60 with an automorphism of order 29 and show that the total number of inequivalent Hadamard matrices with this property is 266.The ternary code and the code over F 5 spanned by the rows of a Hadamard matrix of order 60 are self-dual.A computation of the minimum weights of the ternary codes spanned by the 266 inequivalent Hadamard matrices of order 60 with an automorphism of order 29 shows that only four of these matrices span a ternary extremal self-dual code, and these four matrices appear in the three known inequivalent ternary extremal self-dual codes.Among the remaining codes, several new ternary near-extremal self-dual codes with previously unknown weight enumerators are found.Self-dual codes over F 5 spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are also computed.In addition, new binary near-extremal doubly even self-dual codes of length 120 with previously unknown weight enumerators are constructed from some of the 29,14) designs with an automorphism of order 29.
In Section 4, we give the classification of Hadamard 2-(63, 31, 15) designs with an automorphism of order 31 having one fixed point.There are exactly 826 non-isomorphic designs with this property.Using these designs, we classify the Hadamard matrices of order 64 with an automorphism of order 31, and show that there are 414 inequivalent matrices with this property.The extended code of a binary code of length 63 spanned by the incidence matrix of a 31,15) design is doubly even.Among the extended codes of the 826 non-isomorphic 2-(63, 31, 15) designs with an automorphism of order 31, there are 794 self-dual codes, and among these codes there are 28 inequivalent extremal doubly even self-dual codes of length 64 with an automorphism of order 31.
In Section 5, we give a summary of the classification of Hadamard 2-(2p + 1, p, (p − 1)/2) designs with an automorphism of order p having one fixed point and Hadamard matrices of order 2p + 2 with an automorphism of order p, where p ≤ 31 is an odd prime.
All computer calculations in this paper were done by programs in the language C and programs in Magma [6].

Preliminaries
In this section, we give some definitions and known results about Hadamard matrices, designs and codes used in this paper.We also review a method from [10], [29] and [31] for classifying Hadamard matrices of order 2p + 2 with an automorphism of order p, where p is an odd prime.

Hadamard matrices, designs and codes
Throughout this paper, I n denotes the identity matrix of order n, A T denotes the transpose of a matrix A, and J denotes the all-one matrix of appropriate size.
A Hadamard matrix H of order n is an n×n matrix whose entries are from {1, −1} such that HH T = nI n .It is known that the order n is necessarily 1, 2, or a multiple of 4. Two Hadamard matrices H and K are equivalent if there are (1, −1, 0)-monomial matrices P and Q with K = P HQ.An automorphism of a Hadamard matrix H is an equivalence of H to itself.The set of all automorphisms of H forms a group under composition called the automorphism group Aut(H) of H.For orders up to 32, all inequivalent Hadamard matrices are known (see [18] for order 32).
A t-(v, k, λ) design D is a pair of a set P of v points and a collection B of k-element subsets of P (called blocks) such that every t-element subset of P is contained in exactly λ blocks.Often a t-(v, k, λ) design is simply called a t-design.Two t-(v, k, λ) designs are isomorphic if there is a bijection between their point sets that maps the blocks of the first design into the blocks of the second design.An automorphism of a t-(v, k, λ) design D is any isomorphism of the design with itself and the set consisting of all automorphisms of D is called the automorphism group Aut(D) of D. A t-design can be represented by its (block by point) incidence matrix A = (a ij ), where a ij = 1 if the i-th block contains the j-th point and a ij = 0 otherwise.The complementary (resp.dual ) design of a t-design with incidence matrix A is the design with incidence matrix J − A (resp.A T ).The blocks of a t-(v, k, λ) design D which contain a given point form a (t − 1)-(v − 1, k − 1, λ) design called a derived design of D. A 2-design is called symmetric if the numbers of points and blocks are the same.A symmetric 2-(4t + 3, 2t + 1, t) design is called a Hadamard 2-design.
Let F p = {0, 1, . . ., p − 1} denote the finite field of order p, where p is a prime.An [n, k] code C over F p is a k-dimensional vector subspace of F n p .In this paper, we consider codes over F p (p = 2, 3, 5) only.A code over F 2 and F 3 are called binary and ternary, respectively.The parameters n and k are called the length and dimension of C, respectively.The weight wt(x) of a vector x ∈ F n p is the number of non-zero components of x.A vector of C is called a codeword.The minimum non-zero weight of all codewords in C is called the minimum weight of C. The weight enumerator of C is defined as c∈C y wt(c) .Two codes C and C ′ over F p are equivalent if there is a monomial matrix P over F p with C ′ = C • P , where A binary code C is doubly even if wt(x) ≡ 0 (mod 4) for all codewords x ∈ C. A binary doubly even self-dual code of length n exists if and only if n is divisible by eight.A ternary self-dual code of length n exists if and only if n is divisible by four.It was shown in [21] that the minimum weight d of a binary doubly even self-dual code (resp.ternary self-dual code) of length n is bounded by d ≤ 4⌊n/24⌋ + 4 (resp.d ≤ 3⌊n/12⌋ + 3).If d = 4⌊n/24⌋ + 4 (resp.d = 4⌊n/24⌋), then the binary doubly even self-dual code is called extremal (resp.near-extremal ).If d = 3⌊n/12⌋ + 3 (resp.d = 3⌊n/12⌋), then the ternary self-dual code is called extremal (resp.near-extremal ).

Hadamard matrices of order 2p + 2 with an automorphism of order p
A method for classifying Hadamard matrices of order 2p + 2 with an automorphism of order p, where p is an odd prime with p > 3, is given in [10], [29] and [31].Using this method, a classification of Hadamard matrices of order 2p + 2 with an automorphism of order p was completed in [10], [29] and [31] for p = 13, 17, 19, 23.Here we review the method.Let p > 3 be an odd prime.If a Hadamard matrix H of order 2p + 2 has an automorphism of order p, then H is constructed from a Hadamard 2-(2p + 1, p, (p − 1)/2) design with an automorphism of order p having one fixed point.Note that this follows from a well-known connection between Hadamard matrices and symmetric designs, together with a bound on the number of fixed points [11, p. 82].Let D be a Hadamard 2-(2p+1, p, (p−1)/2) design with an automorphism of order p having one fixed point.Then D has an incidence matrix of the form: where M, N, P and Q are p × p circulant matrices satisfying If the circulant matrices M, N, P and Q satisfy (2), the matrix A in ( 1) is an incidence matrix of a Hadamard 2-(2p + 1, p, (p − 1)/2) design if and only if the following equalities hold: where Then H is a Hadamard matrix of order 2p+2 with an automorphism of order p.
In the next two sections, we use the above method to extend the classification of Hadamard matrices of order 2p + 2 with an automorphism of order p = 29 and p = 31.
3 Hadamard matrices of order 60 with an automorphism of order 29 Using the method given in Section 2.2, in this section, we give the classification of Hadamard 2-(59, 29, 14) designs with an automorphism of order 29 having one fixed point.Using this classification, we give the classification of Hadamard matrices of order 60 with an automorphism of order 29.We construct ternary self-dual codes and self-dual codes over F 5 from the Hadamard matrices of order 60.We also construct binary doubly even self-dual codes from the Hadamard 2-(59, 29, 14) designs.designs with an automorphism of order 29 having one fixed point.In order to classify such 2-designs, we consider incidence matrices of the form (1).By a program implemented in the language C using functions from the GNU Scientific Library (GSL), our exhaustive computer search found all circulant matrices M, N, P and Q satisfying the conditions (2)- (7).In this calculation, we identify 14-elements subsets S M , S Using this classification, all Hadamard matrices of order 60 with an automorphism of order 29 are obtained as matrices of the form ( 8) and ( 9), which need be checked further for equivalence.After equivalence testing, we completed the classification of Hadamard matrices of order 60 with an automorphism of order 29.This was done by using the Magma function IsHadamardEquivalent. Then we have the following: Proposition 1.There are 531 non-isomorphic Hadamard 2-(59, 29, 14) designs with an automorphism of order 29 having one fixed point.There are 266 inequivalent Hadamard matrices of order 60 with an automorphism of order 29.
It is trivial that an automorphism of order 29 of H 60,i induces an automorphism of order 29 of C 3 (H 60,i ).There are three inequivalent ternary extremal self-dual codes of length 60 with an automorphism of order 29 [7].The three codes are the extended quadratic residue code QR 60 , the Pless symmetry code P 60 and the code NV 60 found by Nebe and Villar [24].The codes QR 60 and P 60 contain a type I Paley-Hadamard matrix H P I and a type II Paley-Hadamard matrix H P II , respectively (see [33]).The code NV 60 contains two inequivalent Hadamard matrices H N V 1 and H N V 2 of order 60 [2].From Table 3, we have the following: Proposition 2. Let H be a Hadamard matrix H of order 60 with an automorphism of order 29 such that C 3 (H) generates a ternary extremal self-dual code.Then H is equivalent to one of the four inequivalent Hadamard matrices Table 3: Minimum weights of ternary self-dual codes C 3 (H 60,i ) d i 9 239, 256 12 2, 6, 7, 11, 12, 16, 20, 25, 26, 30, 31, 32, 33, 34, 36, 37, 40, 44, 45, 53 Recently, some restrictions on the weight enumerators of ternary nearextremal self-dual codes of length divisible by 12 were proved in [1], namely, the weight enumerator of a ternary near-extremal self-dual code of length 60 is of the form: where α = 8β with β ∈ {1, 2, . . ., 5148} [1].It is known that there is a ternary near-extremal self-dual code of length 60 having weight enumerator where Γ 60,1 and Γ 60,2 are listed in [1, Tables 22 and 27].It follows from Table 3 We note that no ternary near-extremal codes with weight enumerators given in Proposition 3 were previously known.
, where B 59,i = J − A 59,i is the incidence matrix of the complementary design of D 59,i .In the context of binary codes, we consider the elements 0, 1 of Z as the elements 0, 1 of F 2 , respectively.It is trivial that if D and D ′ are isomorphic Hadamard 2-(59, 29, 14) designs then C 2 (D) ∼ = C 2 (D ′ ).The binary codes C 2 (D 59,i ) are doubly even self-dual codes [30].The minimum weights d of C 2 (D 59,i ) were computed with the Magma function MinimumWeight, and are listed in Table 5.It is trivial that an automorphism of order 29 of D 59,i induces an automorphism of order 29 of C 2 (D 59,i ).Note that there is no binary extremal doubly even self-dual code of length 120 with an automorphism of order 29 [9].Also, it is currently not known whether there is a binary extremal doubly even self-dual code of length 120.From Table 5, the codes C 2 (D 59,i ) (i = 21, 24, 209, 210, 529, 530) are binary near-extremal doubly even self-dual codes of length 120.We verified with the Magma function NumberOfWords that the numbers of codewords of weight 20 in these codes are 71862, 71862, 98484, 98484, 104052 and 104052, respectively.If D 59,i and D 59,j are Hadamard 2-(59, 29, 14) designs which are derived designs of isomorphic 3-(60, 30, 14) designs, then C 2 (D 59,i ) and C 2 (D 59,j ) are equivalent [30,Theorem 2].For (i, j) ∈ { (21,24), (209, 210), (529, 530)}, we verified with the Magma function IsIsomorphic that D 59,i and D 59,j are derived designs of isomorphic 3-(60, 30,14) designs.This implies that C 2 (D 59,i ) and C 2 (D 59,j ) are equivalent.
The weight enumerator of a binary near-extremal doubly even self-dual code of length 120 is of the form:  1].
Remark 4. The number of inequivalent binary near-extremal doubly even self-dual codes of length 120 constructed in [15] was incorrectly reported as 500.We point out that the correct number is 502.Hence, 528 inequivalent binary near-extremal doubly even self-dual codes of length 120 were previously known.

Proposition 5. (i)
There is a binary near-extremal doubly even self-dual code of length 120 with weight enumerator W 2,120 for α = 98484 and 104052.
(ii) There are at least 530 inequivalent binary near-extremal doubly even self-dual codes of length 120.
We note that no binary near-extremal codes with weight enumerators given in Proposition 5 (i) were previously known.
3.4 Self-dual codes C 5 (H 60,i ) over F 5 Let H be a Hadamard matrix of order 60.We denote by C 5 (H) the code over F 5 generated by the rows of H.In the context of codes over F 5 , we consider the elements 0, 1, −1 of Z as the elements 0, 1, 4 of F 5 , respectively.It is trivial that if H and H ′ are equivalent Hadamard matrices of order 60 then C 5 (H) ∼ = C 5 (H ′ ).In addition, C 5 (H) is self-dual [20].The minimum weights d of C 5 (H 60,i ) (i = 1, 2, . . ., 266) were computed with the Magma function MinimumWeight, and are listed in Table 6.Let d(C 5 (H 60,i )) and N(C 5 (H 60,i )) denote the minimum weight and the number of codewords of minimum weight in C 5 (H 60,i ), respectively.For the pair (i, j) such that d(C 5 (H 60,i )) = d(C 5 (H 60,j )) and N(C 5 (H 60,i )) = N(C 5 (H 60,j )), using the Magma function IsIsomorphic, we determined whether C 5 (H 60,i ) ∼ = C 5 (H 60,j ) or not, where the numbers N(C 5 (H 60,i )) were computed with the Magma function NumberOfWords.Then we found equivalent codes C 5 (H 60,238 ) ∼ = C 5 (H 60,257 ) and there no other pair of equivalent codes among C 5 (H 60,i ).
3.1 Hadamard 2-(59, 29, 14) designs D 59,i and Hadamard matrices H 60,i of order 60 As a first step, we completed the classification of Hadamard 2-(59, 29, 14) 29,14)th the supports of the first rows of M, N, P and Q, respectively.In this way, we found all Hadamard 2-(59,29,14)designs with an automorphism of order 29 which need to be checked further for isomorphism.To test an isomorphism of Hadamard 2-(59, 29, 14) designs, we employed the algorithm given in[23,  p. 15, Theorem 1 (b)].This algorithm considers coloured graphs corresponding to29,14)designs.In this calculation, we used nauty software library[22]for coloured graphs isomorphism testing.After isomorphism testing, we completed the classification of Hadamard 2-(59,29,14)designs with an automorphism of order 29 having one fixed point.

Table 1 :
automorphism group orders | Aut(D 59,i )| of D 59,i were computed with the Magma function AutomorphismGroup, and are listed in Table 1.The automorphism group orders | Aut(H 60,i )| of H 60,i were computed with the Magma function HadamardAutomorphismGroup, and are listed in Table 2. Orders of the automorphism groups of D 59,i

Table 2 :
Orders of the automorphism groups of H 60,i Let H be a Hadamard matrix of order 60.We denote by C 3 (H) the ternary code generated by the rows of H.In the context of ternary codes, we consider the elements 0, 1, −1 of Z as the elements 0, 1, 2 of F 3 , respectively.It is trivial that if H and H ′ are equivalent Hadamard matrices of order 60 then C 3 (H) ∼ = C 3 (H ′ ).Since 3 divides 60 and 3 2 does not divide 60, by considering the elementary divisors of H 60,i , it follows that the codes C 3 (H 60,i ) (i = 1, 2, ..., 266) are self-dual (see[19,Section IV]).The minimum weights d of C 3 (H 60,i ) were computed with the Magma function MinimumWeight, and are listed in Table3.Using the Magma function IsIsomorphic, we found the following pairs (i, j) of equivalent codes C 3 (H 60,i ) ∼ = C 3 (H 60,j ): that 154 of the codes C 3 (H 60,i ) are near-extremal.The values α = 8β in the weight enumerators W 3,60 for the 154 near-extremal self-dual codes C 3 (H 60,i ) are listed in Table4.This was calculated by the Magma function NumberOfWords.From Table4, we have the following: Proposition 3.There is a ternary near-extremal self-dual code of length 60 having weight enumerator W 3,60 for α ∈ {8β | β ∈ Γ 60,3 }, where

Table 4 :
Ternary near-extremal self-dual codes of length 36 3.3 Binary doubly even self-dual codes C 2 (D 59,i ) Let A 59,i be the incidence matrix of a Hadamard 2-(59, 29, 14) design D 59,i and let C 2 (D 59,i ) be the binary [120, 60] code generated by the rows of the following matrix:

Table 9 ]
. From Table 6, C 5 (H 60,21 ) and C 5 (H 60,266 ) have the largest minimum weight among currently known self-dual codes of length 60.We verified with the Magma function NumberOfWords that the numbers of codewords of minimum weight in QR 60 , C 5 (H 60,21 ) and C 5 (H 60,266 ) are 410640, 410640 and 288840, respectively.Therefore, there are at least two inequivalent self-dual codes over F 5 of length 60 and minimum weight 18.

Table 6 :
[3]imum weights of self-dual codes C 5 (H 60,i ) over F 5 4 Hadamard matrices of order 64 with an automorphism of order 31 Hadamard 2-(63, 31, 15) designs D 63,i and Hadamard matrices H 64,i of order 64The approach used in the classification is similar to that given in the previous section, so in this section, only results are given.Hence, the largest prime which can divide the order of the automorphism group of a Hadamard matrix of order 64 is 31.The incidence matrices of the above 826 non-isomorphic Hadamard 2-(63, 31, 15) designs D 63,i (i = 1, 2, ..., 826) and the above 414 inequivalent Hadamard matrices H 64,i (i = 1, 2, ..., 414) from Proposition 6 can be obtained electronically from[3].The automorphism group orders | Aut(D 63,i )| of D 63,i and the automorphism group orders | Aut(H 64,i )| of H 64,i are listed in Tables7 and 8, respectively.
Proposition 6.There are 826 non-isomorphic Hadamard 2-(63, 31, 15) designs with an automorphism of order 31 having one fixed point.There are 414 inequivalent Hadamard matrices of order 64 with an automorphism of order 31.Remark 7. If p is an odd prime dividing the order of the automorphism group of a Hadamard matrix of order n ≥ 4, then either p divides n or n − 1, or p ≤ n/2 − 1 [29].

Table 7 :
Orders of the automorphism groups of D 63,i

Table 8 :
31,15) of the automorphism groups of H 64,iIt is trivial that if D and D ′ are isomorphic31,15)designs then C ′ 2 (D) ∼ = C ′ 2 (D ′ ).In addition, it is trivial that C ′ 2 (D 63,i) are doubly even.The dimensions dim(C ′ 2 (D 63,i )) were computed with the Magma function Dimension, and are listed in Table9.The minimum weights d of the 794 doubly even self-dual codes C ′ 2 (D 63,i ) were computed with the Magma function MinimumWeight, and are listed in Table

Table 9 :
Dimensions of C ′ 2 (D 63,i ) We end this paper by listing a summary of the classification of Hadamard 2-(2p + 1, p, (p − 1)/2) designs with an automorphism of order p having one fixed point and the classification of Hadamard matrices of order 2p + 2 with an automorphism of order p for p ≤ 31.The number N(D 2p+1 ) of nonisomorphic Hadamard 2-(2p + 1, p, (p − 1)/2) designs with an automorphism of order p having one fixed point and the number N(H 2p+2 ) of inequivalent Hadamard matrices of order 2p + 2 with an automorphism of order p are listed in Table11, along with relevant references.

Table 10 :
Minimum weights of doubly even self-dual codes C ′ 2 (D 63,i )