Extremal Self-dual Codes Of Length Research Articles

Hadamard matrices related to a certain series of ternary self-dual codes

In 2013, Nebe and Villar gave a series of ternary self-dual codes of length $$2(p+1)$$ for a prime p congruent to 5 modulo 8. As a consequence, the third ternary extremal self-dual code of length 60 was found. We show that these ternary self-dual codes contain codewords which form a Hadamard matrix of order $$2(p+1)$$ when p is congruent to 5 modulo 24. In addition, we show that the ternary self-dual codes found by Nebe and Villar are generated by the rows of the Hadamard matrices. We also demonstrate that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices.

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.

Quadruple bordered constructions of self-dual codes from group rings

In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring $\mathbb {F}_{4}+u\mathbb {F}_{4}$. We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.

Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F2+uF2, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.

Binary extremal self-dual codes of length 60 and related codes

We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual $[60,30,12]$ codes, we also construct extremal singly even self-dual $[58,29,10]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight $1$.

On the performance of optimal double circulant even codes

In this note, we investigate the performance of optimal double circulant even codes which are not self-dual, as measured by the decoding error probability in bounded distance decoding. To achieve this, we classify the optimal double circulant even codes that are not self-dual which have the smallest weight distribution for lengths up to 72. We also give some restrictions on the weight distributions of (extremal) self-dual [54, 27, 10] codes with shadows of minimum weight 3. Finally, we consider the performance of extremal self-dual codes of lengths 88 and 112.

A projection decoding of a binary extremal self-dual code of length 40

As far as we know, there is no decoding algorithm of any binary self-dual [40, 20, 8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40, 20, 8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two new efficient decoding algorithms for an extremal binary doubly-even self-dual [40, 20, 8] code $$C_{40,1}^{DE}$$C40,1DE by hand with the help of a Hermitian self-dual [10, 5, 4] code $$E_{10}$$E10 over GF(4). The main idea of this decoding is to project codewords of $$C_{40,1}^{DE}$$C40,1DE onto $$E_{10}$$E10 so that it reduces the complexity of the decoding of $$C_{40,1}^{DE}$$C40,1DE. The first algorithm is called the representation decoding algorithm. It is based on the pattern of codewords of $$E_{10}$$E10. Using certain automorphisms of $$E_{10}$$E10, we show that only eight types of codewords of $$E_{10}$$E10 can produce all the codewords of $$E_{10}$$E10. The second algorithm is called the syndrome decoding algorithm based on $$E_{10}$$E10. It first solves the syndrome equation in $$E_{10}$$E10 and finds a corresponding binary codeword of $$C_{40,1}^{DE}$$C40,1DE.

Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over F2+uF2

A classification of all four-circulant extremal codes of length 32 over F2 + uF2 is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with ?=80 in W64,2 is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators are constructed for the first time. These codes have the values ?=1, 30, 34, 84, 94 in W66,1.

On extremal self-dual codes of length 120

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even $$[120, 60, 24]$$ [ 120 , 60 , 24 ] code are $$2, 3, 5, 7, 19, 23$$ 2 , 3 , 5 , 7 , 19 , 23 and $$29$$ 29 . Furthermore we prove that automorphisms of prime order $$p \ge 5$$ p ? 5 have a unique cycle structure. Parts of the results are based on computer computations.

Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that gp is a fixed point free involution. If C is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(gp) which consists of the set of fixed points of gp, we prove that C is a projective F2〈g 〉-module if and only if a natural projection of C(gp) is a self-dual code. We then discuss easy-to-handle criteria to decide if C is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.

A Complete Classification of Doubly Even Self-dual Codes of Length 40

A complete classification of binary doubly even self-dual codes of length $40$ is given. As a consequence, a classification of binary extremal self-dual codes of length $38$ is also given.

Classification of Extremal and $s$-Extremal Binary Self-Dual Codes of Length 38

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal self-dual codes, two s-extremal codes, and 1730 s-extremal codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of -extremal codes permits to achieve the classification of all -extremal codes with .

On Extremal Self-Dual Codes of Length 96

Let C be a binary extremal self-dual code of length 96. We prove that an automorphism of C of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then Aut(C) is solvable and its order divides 253 or 255 or Aut(C) is the alternating group A5 which is the only possible group of order 60. Furthermore, |Aut(C)| = 20 or 40 cannot occur.

The binary extremal self-dual codes of lengths 38 and 40

We classify the extremal self-dual codes of lengths 38 or 40 having an automorphism of order 3 with six independent 3-cycles, 10 independent 3-cycles, or 12 independent 3-cycles. In this way we complete the classification of binary extremal self-dual codes of length up to 48 having automorphism of odd prime order.

On the classication of binary self-dual [44, 22, 8] codes with an automorphism of order 3 or 7

All binary self-dual [44, 22, 8], codes with an automorphism of order 3 or 7 are classified. In this way, we complete the classification of extremal self-dual codes of length 44 having an automorphism of odd prime order.

Hadamard matrices of order 32 and extremal ternary self-dual codes

A ternary self-dual code can be constructed from a Hadamard matrix of order congruent to 8 modulo 12. In this paper, we show that the Paley---Hadamard matrix is the only Hadamard matrix of order 32 which gives an extremal self-dual code of length 64. This gives a coding theoretic characterization of the Paley---Hadamard matrix of order 32.

Automorphisms of Extremal Self-Dual Codes

Let C be a binary extremal self-dual code of length n ? 48. We prove that for each ? ? Aut(C) of prime order p ? 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of p-cycles. It turns out that large primes p, i.e., n-p small, seem to occur in |Aut(C)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of length n = p + 1 are extremal only in the cases n =8, 24, 32, 48, 80, and 104.

Classification of ternary extremal self-dual codes of length 28

All 28-dimensional unimodular lattices with minimum norm 3 are known. Using this classification, we give a classification of ternary extremal self-dual codes of length 28. Up to equivalence, there are 6,931 such codes.

Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices

In this paper, a construction of ternary self-dual codes based on negacirculant matrices is given. As an application, we construct new extremal ternary self-dual codes of lengths 32, 40, 44, 52 and 56. Our approach regenerates all the known extremal self-dual codes of lengths 36, 48, 52 and 64. New extremal ternary quasi-twisted self-dual codes are also constructed.

Extremal self-dual codes of length 64 through neighbors and covering radii

We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.