Codes over Rk, Gray maps and their binary images

Linear complementary dual (LCD) codes 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 Rk. 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.

The combinatorics of LCD codes: linear programming bound and orthogonal matrices

A class of constacyclic codes over Fp+vFp and its Gray image

In this work, we investigate linear codes over the ring $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43---65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32---45, 1999). We then characterize the $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$ -linear codes. The duals and the complete weight enumerators for $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over $${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$$ are obtained.

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this paper, we investigate the structure and properties of cyclic codes over the ring F 2+vF 2 . We first study the relationship between cyclic codes over F 2+vF 2 and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length n over F 2+vF 2.

Some Results on Cyclic Codes Over ${F}_{2}+v{F}_{2}$

In this work, we investigate linear codes over the ring [FORMULA] . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the [FORMULA] -linearity of binary codes under the Gray map and give a main class of binary codes as an example of [FORMULA] -linear codes. The duals and the complete weight enumerators for [FORMULA] -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over [FORMULA] are obtained.

Linear codes over F 2 + uF 2 + vF 2 + uvF 2 .

Linear codes over Z4+uZ4: MacWilliams identities, projections, and formally self-dual codes

Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z_4+uZ_4 and their Z_4-images together with some good examples of formally self-dual Z_4-codes obtained through these constructions.

Bahattin Yildiz

Papers

Codes over Rk, Gray maps and their binary images

Linear codes over F 2 + uF 2 + vF 2 + uvF 2 .

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

Linear codes over Z4+uZ4: MacWilliams identities, projections, and formally self-dual codes

Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes