Binary Self-dual Codes Research Articles

New extremal binary self-dual codes of length 68

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F_2 + uF_2 + u^2F_2, where u^3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with beta=117, 120, 133 in W68;1 and with gamma = 1, beta=49, 57, 59 and codes with gamma=2, beta=69, 81 in W68;2 are constructed for the ?first time in the literature. In addition to these, some known such codes are reconstructed via this extension. The results are tabulated.

Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7

This paper studies and classifies all binary self-dual $[62, 31, 12]$ and $[64, 32, 12]$ codes havingan automorphism of order 7 with 8 cycles. This classification is done by applying a method forconstructing binary self-dual codes with an automorphism of odd prime order $p$.There are exactly 8 inequivalent binary self-dual $[62, 31, 12]$ codes with an automorphism oftype $7-(8,6)$. As for binary $[64,32,12]$ self-dual codes with an automorphism of type $7-(8,8)$ thereare 44465 doubly-even and 557 singly-even such codes. Some of the constructed singly-even codes for both lengthshave weight enumerators for which the existence was not known before.

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.

Constructing formally self-dual codes over [formula omitted

In this work, we study construction techniques of formally self-dual codes over the infinite family of rings Rk=F2[u1,u2,…,uk]/〈ui2=0,uiuj=ujui〉. These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters [72,36,14], [56,28,12], [44,22,10] and odd formally self-dual binary codes of parameters [72,36,13], all of which have better minimum distances than the best known self-dual codes of the same lengths.

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.

The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$

A computer calculation with $M$AGMA shows that there is noextremal self-dual binary code $\mathcal{C}$ of length $72$ whoseautomorphism group contains the symmetric group of degree $3$, thealternating group of degree $4$ or the dihedral group of order $8$.Combining this with the known results in the literature one obtainsthat $Aut(\mathcal{C})$ has order at most $5$ or isisomorphic to the elementary abelian group of order $8$.

Classification of self-dual codes of length 50 with an automorphism of odd prime order

By applying a method for constructing binary self-dual codes with an automorphism of odd prime order $$p$$ p , we give a full classification of all optimal binary self-dual codes of length 50 having an automorphism of order 3. As a consequence, we give a full classification of all $$[50, 25, 10]$$ [ 50 , 25 , 10 ] codes possessing an automorphism of odd prime order. Up to equivalence, there are exactly 177,601 such codes. This completely determines all possibilities for the cardinality of the automorphism group of such a code. Also, we show that there are at least 52 non-isomorphic quasi-symmetric 2-(49, 9, 6) designs, derived from the $$[50,25,10]$$ [ 50 , 25 , 10 ] codes with a particular weight enumerator.

The automorphism group of a self-dual [formula omitted] code is not an elementary abelian group of order 8

The existence of an extremal self-dual binary linear code C of length 72 is a long-standing open problem. We continue the investigation of its automorphism group: looking at the combination of the subcodes fixed by different involutions and doing a computer calculation with Magma , we prove that Aut ( C ) is not isomorphic to the elementary abelian group of order 8. Combining this with the known results in the literature one obtains that Aut ( C ) has order at most 5.

New binary self-dual codes of lengths 50–60

Using a method for constructing binary self-dual codes with an automorphism of odd prime order $$p$$ p , we give a full classification of all optimal binary self-dual $$[50+2t,25+t]$$ [ 50 + 2 t , 25 + t ] codes having an automorphism of order 5 for $$t=0,\dots ,5$$ t = 0 , ? , 5 . As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual $$[52, 26, 10]$$ [ 52 , 26 , 10 ] code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3,000 new doubly-even $$[56,28,12]$$ [ 56 , 28 , 12 ] self-dual codes.

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.

New extremal binary self-dual codes of length 66 as extensions of self-dual codes over Rk

We obtain extremal binary self-dual codes of parameters [64,32,12] as binary images of self-dual codes over R1, R2 and R3 by employing different methods. We then apply the extension theorem to these codes to obtain a number of extremal binary self-dual codes of length 66 with trivial automorphism groups. Fifteen of the codes we obtain have new β values in W66,3, of which only three were known to exist before. We also find nine codes with new β values in W66,1, thus updating the list of such known codes.

New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes

A lift of binary self-dual codes to the ring $R_2$ is described. By using this lift, a family of self-dual codes over $R_2$ of length $17$ are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length $68$ are obtained. For the first time in the literature, extremal binary codes of length $68$ with $\gamma=4$ and $\gamma = 6$ in $W_{68,2}$ have been obtained. In addition to these, six new codes with $\gamma = 0$ and fourteen new codes with $\gamma = 2$ in $W_{68,2}$ have also been found.

A new class of majority-logic decodable codes derived from polarity designs

The polarity designs, introduced in [9], are combinatorial 2-designs having the same parameters as a projective geometry design $PG_s(2s,q)$ formed by the $s$-subspaces of $PG(2s,q)$, $s\ge 2$, $q=p^t$, $p$ prime. If $q=p$ is a prime, a polarity design has also the same $p$-rank as $PG_s(2s,p)$. If $q=2$, any polarity 2-design is extendable to a 3-design having the same parameters and 2-rank as an affine geometry design $AG_{s+1}(2s+1,2)$ formed by the $(s+1)$-subspaces of $AG(2s+1,2)$. It is shown in this paper that a linear code being the null space of the incidence matrix of a polarity design can correct by majority-logic decoding the same number of errors as the projective geometry code based on $PG_s(2s,q)$. In the binary case, any polarity 3-design yields a binary self-dual code with the same parameters, minimum distance, and correcting the same number of errors by majority-logic decoding as the Reed-Muller code of length $2^{2s+1}$ and order $s$.

The Automorphism Group of an Extremal $[{72,36,16}]$ Code Does Not Contain $Z_{7}$, $Z_{3}\times Z_{3}$, or $D_{10}$

A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 that has an automorphism group containing either the dihedral group of order 10, the elementary abelian group of order 9, or the cyclic group of order 7. Combining this with the known results in the literature, one obtains that the order of Aut(C) is either 5 or divides 24.

New extremal binary self-dual codes of length 58 as R3-lifts from the shortened binary [formula omitted] Hamming code

Abstract In this correspondence, a lift of the shortened binary [ 8 , 4 , 4 ] Hamming code to the ring R3 is described. Using this lift, a family of self-dual codes over R3 of length 7 are obtained. By taking the binary images of these codes and applying the extension method given in [15] , 11 new extremal binary self-dual [ 58 , 29 , 10 ] codes with new weight enumerators are found which were not previously known to exist.

On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly for length n = 1024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.

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 .

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over \(\mathbb{F}_{2}\), \(\mathbb{F}_{3}\), \(\mathbb{F}_{4}\), \(\mathbb{F}_{5}\), and \(\mathbb{F}_{7}\). In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.

An Algorithm for Classification of Binary Self-Dual Codes

An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.

Every binary self-dual code arises from Hilbert symbols

In this paper we construct binary self-dual codes using the \'etale cohomology of $\mathbb{Z}/2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all self-dual codes of length at least 4 arise from Hilbert pairings on rings of $S$-integers of $\Q$. This is an arithmetic counterpart of a result of Kreck and Puppe, who used cobordism theory to show that all self-dual codes arise from Poincar\'e duality on real three manifolds.