Extremal Self-dual Codes 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.

Ternary quantum codes constructed from extremal self-dual codes and self-orthogonal codes

<abstract><p>In this paper, we consider $ S $-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new $ S $-chain search method. As computational results, 18 $ S $-chains with large distances are obtained, and many good quantum codes can be derived from those $ S $-chains by Steane construction, some of which improve the previous results.</p></abstract>

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.

On the classification of binary self-dual codes admitting imprimitive rank 3 permutation groups

One of the questions of current interest in coding theory is the following: given a finite non-solvable permutation group G acting transitively on a set $$\Omega $$ , under what conditions on G are self-dual codes invariant under G existent or nonexistent? In this paper, this problem is investigated under the hypothesis that the group G is an imprimitive rank 3 permutation group. It is proven that if G is an imprimitive rank 3 permutation group acting transitively on the coordinate positions of a self-dual binary code C then G is one of $${\mathrm{M}}_{11}$$ of degree 22; $${\mathrm{Aut}}({\mathrm{M}}_{12})$$ of degree 24; $$\mathrm{PSL}(2,q)$$ of degree $$2(q +1)$$ for $$q {\equiv 1}{({\mathrm{mod}}\,4)};$$ $$\mathrm{PSL}(m, q)$$ of degree $$2\times \frac{q^m-1}{q-1}$$ for $$m \ge 3$$ odd and q an odd prime; $$\mathrm{PSL}(m, q)$$ of degree $$2\times \frac{q^m-1}{q-1}$$ for $$m \ge 4$$ even and q an odd prime, and $$\mathrm{PSL}(3,2)$$ of degree 14. When combined with a result on the classification of binary self-dual codes invariant under primitive rank 3 permutation groups of almost simple type this yields a result on the non-existence of extremal binary self-dual codes invariant under quasiprimitive rank 3 permutation groups of almost simple type.

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 $1$.

Performance of ternary double circulant, double twistulant, and self-dual codes

We study the performance of ternary isodual codes which are not self-dual and ternary self-dual codes, as measured by the decoding error probability with bounded distance decoding. We compare the performance of ternary double circulant and double twistulant codes which are not self-dual with ternary extremal self-dual codes. We also investigate the performance of ternary self-dual codes having large minimum weights.

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.

Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes

A generalized hypercube graph $\Q_n(S)$ has $\F_{2}^{n}=\{0,1\}^n$ as the vertex set and two vertices being adjacent whenever their mutual Hamming distance belongs to $S$, where $n \ge 1$ and $S\subseteq \{1,2,\ldots, n\}$. The graph $\Q_n(\{1\})$ is the $n$-cube, usually denoted by $\Q_n$. We study graph boolean products $G_1 = \Q_n(S)\times \Q_1, G_2 = \Q_{n}(S)\wedge \Q_1$, $G_3 = \Q_{n}(S)[\Q_1]$ and show that binary codes from neighborhood designs of $G_1, G_2$ and $G_3$ are self-orthogonal for all choices of $n$ and $S$. More over, we show that the class of codes $C_1$ are self-dual. Further we find subgroups of the automorphism group of these graphs and use these subgroups to obtain PD-sets for permutation decoding. As an example we find a full error-correcting PD set for the binary $[32, 16, 8]$ extremal self-dual code.

Construction of extremal self-dual codes over $${\mathbb {Z}}_{8}$$ Z 8 and $${\mathbb {Z}}_{16}$$ Z 16

We present a method of constructing free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over $${\mathbb {Z}}_{2^m}$$Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer $$m \ge 2$$mź2. We find explicit algorithms for construction of self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over $${\mathbb {Z}}_{2^m}$$Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 up to lengths 40.

New extremal binary self-dual codes from [formula omitted]-lifts of quadratic circulant codes over [formula omitted

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F4+uF4 as well as to F4, and as a result, extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension theorems as well as the ring extension version are used to obtain seven extremal self-dual binary codes of length 58, twenty-four extremal self-dual binary codes of length 66 and twenty-nine extremal self-dual binary codes of length 68, all with new weight enumerators, updating the list of all the known extremal self-dual codes in the literature.

New binary quantum stabilizer codes from the binary extremal self-dual $$[48, 24, 12]$$ [ 48 , 24 , 12 ] code

This paper is devoted to constructing binary quantum stabilizer codes based on the binary extremal self-dual code of parameters $$[48, 24, 12]$$[48,24,12] by Steane's construction. First, we provide an explicit generator matrix for the unique self-dual $$[48,24,12]$$[48,24,12] code to see it as a one-generator quasi-cyclic one and obtain six optimal self-orthogonal codes of parameters $$[48 - t, 24 - t, 12]$$[48-t,24-t,12] for $$1 \le t \le 6$$1≤t≤6 with dual distances from 11 to 7 by puncturing the $$[48,24,12]$$[48,24,12] code. Second, a special type of subcode structures for self-orthogonal codes is investigated, and then ten derived dual chains are designed. Third, twelve binary quantum codes are constructed from these derived dual pairs within dual chains using Steane's construction. Ten of them, $$[[42,10,8]], [[44,11,8]], [[45,10,8]], [[45,8,9]], [[46,12,8]]$$[[42,10,8]],[[44,11,8]],[[45,10,8]],[[45,8,9]],[[46,12,8]], $$[[46,9,9]], [[46,5,10]], [[48,13,8]], [[48,9,9]]$$[[46,9,9]],[[46,5,10]],[[48,13,8]],[[48,9,9]], and $$[[48,4,11]]$$[[48,4,11]], achieve as good parameters as the best known ones with comparable lengths and dimensions. Two other codes of parameters $$[[47,4,11]]$$[[47,4,11]] and $$[[48,3,12]]$$[[48,3,12]] are record breaking in the sense that they improve on the best known ones with the same lengths and dimensions in terms of distance.

On involutions in extremal self-dual codes and the dual distance of semi self-dual codes

A classical result of Conway and Pless is that a natural projection of the fixed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual [13]. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes.In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes.As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and σ∈Aut(C) be a fixed point free automorphism of order 2. If m is odd or if m=2k with (5k−1k−1) odd then C is a free F2〈σ〉-module. This result has quite strong consequences on the structure of the automorphism group of such codes.

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.

New extremal binary self-dual codes of length 64 from $$R_3$$ R 3 -lifts of the extended binary Hamming code

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to $$R_k$$ R k . Using this method we construct self-dual codes over $$R_3$$ R 3 of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have $$\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41$$ β = 1 , 5 , 13 , 17 , 21 , 25 , 29 , 33 , 41 and 52 in $$W_{64,2}$$ W 64 , 2 and they all have automorphism groups of size 8.

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.

Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes

It is known that extremal doubly-even self-dual codes of length $$n\equiv 8$$ n ? 8 or $$0\ (\mathrm {mod}\ 24)$$ 0 ( mod 24 ) yield 3- or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) $$\in \{(32,8,5), (56,12,9), (56,16,9), (56,24,9), (80,16,52)\}$$ ? { ( 32 , 8 , 5 ) , ( 56 , 12 , 9 ) , ( 56 , 16 , 9 ) , ( 56 , 24 , 9 ) , ( 80 , 16 , 52 ) } . With the aid of computer, we obtain 22 generator matrices of bordered double circulant doubly-even self-dual codes of length 48, which enable us to get 506 mutually disjoint 5-(48, k, $$\lambda $$ ? ) designs for (k, $$\lambda $$ ? )=(12, 8),(16, 1356),(20, 36176). Moreover, this implies 5-(48, k; 506)-SEEDs for $$k=12, 16, 20, 24$$ k = 12 , 16 , 20 , 24 .