New extremal binary self-dual codes of lengths 64 and 66 from bicubic planar graphs

In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are tabulated.Comment: 10 pages, 4 table

The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.Comment: submitted to Linear Algebra and Applications on June, 1, 201

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 \lambda-circulant and \lambda-reverse circulant matrices. By using the constructions on $F_2$, we obtain new binary codes of lengths 64 and 68. We also apply the constructions to the ring $R_2$ and considering the $F_2$ and $R_1$-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, 15 new codes of length 66 and 22 new codes of length 68. These codes all have weight enumerators with parameters that were not known to exist in the literature.Comment: 7 table

Codes over Affine Algebras with a Finite Commutative Chain coefficient Ring

We consider codes defined over an affine algebra $\mathcal A=R[X_1,\dots,X_r]/\left\langle t_1(X_1),\dots,t_r(X_r)\right\rangle$, where $t_i(X_i)$ is a monic univariate polynomial over a finite commutative chain ring $R$. Namely, we study the $\mathcal A-$submodules of $\mathcal A^l$ ($l\in \mathbb{N}$). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. {Some codes over Frobenius local rings that are not chain rings are also of this type}. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.Comment: Submitted to Finite Fields and Their Application

The homogeneous weight for RkR_k, related Gray map and new binary quasicyclic codes

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family $R_k$, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over $R_k$ under this Gray map. We then discuss quasitwisted codes over $R_k$ and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasicyclic. In particular, we find a substantial number of optimal binary codes that are quasicyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasicyclic codes kept by Chen.Comment: Submitted to be publishe

ΘS−\Theta_S-cyclic codes over AkA_k

We study $\Theta_S-$cyclic codes over the family of rings $A_k.$ We characterize $\Theta_S-$cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is $\Theta_S-$cyclic then its Hermitian dual is also $\Theta_S-$cyclic. Finally, we give constructions of $\Theta_S-$cyclic codes.Comment: 23 page

Simplex and MacDonald Codes over RqR_{q}

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring $R_{q}=\mathbb{F}_{2}[u_{1},u_{2},\ldots,u_{q}]/\left\langle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle$ for $q \geq 2$. We also consider the construction of simplex and MacDonald codes of types $\alpha$ and $\beta$ over this ring

Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring $RG$ and as such must have an automorphism group that contains $G$ as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative $[72,36,16]$ Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48

On codes over R_{k,m} and constructions for new binary self-dual codes

In this work, we study codes over the ring R_{k,m}=F_2[u,v]/, 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^{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].Comment: 17 page

MacWilliams Type identities for mm-spotty Rosenbloom-Tsfasman weight enumerators over finite commutative Frobenius rings

The $m$-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). $m$-spotty byte error control codes are very suitable for burst correction. M. \"{O}zen and V. Siap [7] proved a MacWilliams identity for the $m$-spotty Rosenbloom-Tsfasman (shortly RT) weight enumerators of binary codes. The main purpose of this paper is to present the MacWilliams type identities for $m$-spotty RT weight enumerators of linear codes over finite commutative Frobenius rings.Comment: Research article, orignial manuscript under review since 2nd November 2012. 9 pages, 4 Tables. arXiv admin note: substantial text overlap with arXiv:1307.1786, arXiv:1307.222