Binary Self-dual Codes Research Articles

Construction of isodual codes over GF(q)

We develop a construction method of isodual codes over GF(q), where q is a prime power; we construct isodual codes over GF(q) of length 2n+2 from isodual codes over GF(q) of length 2n. Using this method, we find some isodual codes over GF(q), where q=2,3 and 5. In more detail, we obtain binary isodual codes of lengths 32, 34, 36, 38, and 40, where all these codes of lengths 32, 34, and 36 are optimal and some codes of length 38 are optimal. We note that all these binary isodual codes are not self-dual codes, and in particular, in the case of length 38 all their weight enumerators are different from those of binary self-dual codes of the same length; in fact, four binary isodual codes of length 38 are formally self-dual even codes. We construct isodual codes over GF(3) and GF(5) of lengths 4, 6, and 8 as well.

NEW EXTREMAL BINARY SELF-DUAL CODES OF LENGTHS 66 AND 68 FROM CODES OVER Rk,m

In this work, four circulant and quadratic double circulant (QDC) constructions are applied to the family of the rings R_k,m. Self-dual binary codes are obtained as the Gray images of self-dual QDC codes over R_k,m. Extremal binary self-dual codes of length 64 are obtained as Gray images of ?-four circulant codes over R_2,1 and R_2,2. Extremal binary self-dual codes of lengths 66 and 68 are constructed by applying extension theorems to the F_2 and R_2,1 images of these codes. More precisely, 11 new codes of length 66 and 39 new codes of length 68 are discovered. The codes with these weight enumerators are constructed for the first time in literature. The results are tabulated.

On codes over 𝓡 k, m and constructions for new binary self-dual codes

Abstract In this work, we study codes over the ring 𝓡 k, m = 𝔽2[u, v]/ 〈uk , vm , uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F 2 k m $\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

On the duality and the direction of polycyclic codes

Polycyclic codes are ideals in quotients of polynomial rings by a principal ideal. Special cases are cyclic and constacyclic codes.A MacWilliams relation between such a code and its annihilator ideal is derived. An infinite family of binary self-dual codes that are also formally self-dualin the classical sense is exhibited.We show that right polycyclic codes are left polycyclic codes with different(explicit) associate vectors and characterize the casewhen a code is both left and right polycyclic for the same associate polynomial. A similar study is led for sequential codes.

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.

New extremal binary self-dual codes from a modified four circulant construction

In this work, we propose a modified four circulant construction for self-dual codes and a bordered version of the construction using the properties of λ-circulant and λ-reverse circulant matrices. By using the constructions on F2, we obtain new binary codes of lengths 64 and 68. We also apply the constructions to the ring R2 and considering the F2 and R1-extensions, we obtain new singly-even extremal binary self-dual codes of lengths 66 and 68. More precisely, we find 3 new codes of length 64, 13 new codes of length 66 and 21 new codes of length 68. These codes all have weight enumerators with parameters that were not known to exist in the literature.

On the Existence of Certain Optimal Self-Dual Codes with Lengths Between 74 and 116

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.

Various constructions for self-dual codes over rings and new binary self-dual codes

In this work, extension theorems are used for self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F2m+uF2m for m=1,2. The duality and distance preserving Gray maps from F4+uF4 to (F2+uF2)2 and F24 are used to obtain self-dual codes whose binary Gray images are [64,32,12]-extremal self-dual. An F2+uF2-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ=3 and many codes with the rare γ=4,6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96,48,16]-codes with new weight enumerators are obtained from four-circulant codes over F4+uF4. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F2+uF2-lifts of binary four circulant codes and thus a lower bound on the number of non-isomorphic 3-(80, 16, 665) designs is modified.

Classification of Binary Self-Dual Codes of Length 40

An efficient isomorph-free generation algorithm for classification of binary self-dual codes with the minimum distance $d = 4$ is presented. It is combined with another isomorph-free generation algorithm for classification of self-dual codes with the minimum distance $d \ge 6$ . A complete classification of all binary self-dual codes of length 40 is given as a result.

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.

AN EFFICIENT CONSTRUCTION OF SELF-DUAL CODES

Abstract. Self-dual codes have been actively studied because of theirconnections with other mathematical areas including t-designs, invari-ant theory, group theory, lattices, and modular forms. We presentedthe building-up construction for self-dual codes over GF(q) with q ≡ 1(mod 4), and over other certain rings (see [19], [20]). Since then, the ex-istence of the building-up construction for the open case over GF(q) withq= p r ≡3 (mod 4) with an odd prime p satisfying p≡3 (mod 4) withr odd has not been solved. In this paper, we answer it positively by pre-senting the building-up construction explicitly. As examples, we presentnew optimal self-dual [16,8,7] codes over GF(7) and new self-dual codesover GF(7) with the best known parameters [24,12,9]. 1. IntroductionSince the development of Algebraic Coding Theory, self-dual codes havebecome one of the main research topics because of their connections withgroups, combinatorial t-designs, lattices, and modular forms (see [26]). Somewell known constructions of self-dual codes include the gluing vector technique([23, 24]) and automorphism group method [15].A recently developed and popular construction is to obtain self-dual codesfrom self-dual codes ofsmaller lengths. In [4, 6], the authors used shadowcodes.Motivated by Harada’s work [12], the second author Kim [17] introduced theso-called building-up construction for binary self-dual codes. It shows that anybinary self-dual code can be built from a self-dual code of a smaller length.Then later, the building-up construction for self-dual codes over finite fieldsGF(q) was developed when q is a power of 2 or q ≡ 1 (mod 4) [19], and thenover finite ring Z

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.

Self-dual [formula omitted] lifts of binary self-dual codes

In this paper, we give a method to lift binary self-dual codes to the ring Rk. The lifting method requires solving a system of linear equations over Rk. This technique is applied to [14,7,4] binary self-dual code to obtain self-dual codes over R2. As Gray images of these codes, a substantial number of [56,28,10] self-dual codes are generated. By using the extension theorem given by Bouyuklieva and Bouyukliev in [2], ten new extremal binary self-dual codes of length 58 with new enumerators are found which were not previously known to exist.

Self-Dual Codes With an Automorphism of Order 11

Using a method for constructing self-dual codes having an automorphism of odd prime order, we classify up to equivalence all binary self-dual codes with an automorphism of order 11 with 6 cycles and minimum distance 12. This classification gives new [72, 36, 12] codes with weight enumerator that was previously not obtained as well as many [66, 33, 12], [68, 34, 12], and [70, 35, 12] codes with new values of the parameters in their respective weight enumerators.

On the automorphisms of order 15 for a binary self-dual $$[96, 48, 20]$$ [ 96 , 48 , 20 ] code

The structure of the binary self-dual codes invariant under the action of a cyclic group of order \(pq\) for odd primes \(p\ne q\) is considered. As an application we prove the nonexistence of an extremal self-dual \([96, 48, 20]\) code with an automorphism of order \(15\) which closes a gap in de la Cruz and Willems (IEEE Trans Inf Theory 57:6820–6823, 2011).

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.

Self-dual codes from 3-class association schemes

3-Class association schemes are used to construct binary self-dual codes. We use the pure and bordered construction to get self-dual codes starting from the adjacency matrices of symmetric and non-symmetric 3-class association schemes. In some specific cases, we also study constructions of self-dual codes over $${\mathbb {Z}}_k$$ . For symmetric 3-class association schemes, we focus on the rectangular scheme and association schemes derived from symmetric designs.

SELF-DUAL CODES AND FIXED-POINT-FREE PERMUTATIONS OF ORDER 2

We construct new binary optimal self-dual codes of length 50. We develop a construction method for binary self-dual codes with a fixed-point-free automorphism of order 2. Using this method, we find new binary optimal self-dual codes of length 52. From these codes, we obtain Lee-optimal self-dual codes over the ring <TEX>$\mathbb{F}_2+u\mathbb{F}_2$</TEX> of lengths 25 and 26.

The Automorphism Group of a Self-Dual &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;$[{72,36,16}]$ &lt;/tex-math&gt;&lt;/inline-formula&gt; Code Does Not Contain &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;$Z_{4}$ &lt;/tex-math&gt;&lt;/inline-formula&gt;

It has been proven in a series of works that the order of the automorphism group of a self-dual binary [72,36,16] code does not exceed five. Up to equivalence, we obtain a parametrization of all self-dual binary codes of length 72 with automorphism of order four, which can be extremal. We use extensive computations in Magma and on a supercomputer to show that the automorphism group of an extremal self-dual binary code of length 72 does not have an element of order four.

New extremal binary self-dual codes of length 68 from quadratic residue codes over [formula omitted

In this work, quadratic residue codes over the ring F2+uF2+u2F2 with u3=u are considered. A duality and distance preserving Gray map from F2+uF2+u2F2 to F23 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self-dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [72,36,12], a Type II [72,36,12] code and a Type II [96,48,16] code with new weight enumerators are obtained through these constructions. The results are tabulated.