On Codes over F q +vF q +v 2 F q

Journal of Applied Mathematics and Computing, 2017

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.

arXiv (Cornell University), 2017

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.

2015

In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.

Involve, a Journal of Mathematics, 2009

New metrics and distances for linear codes over the ring Fq[u]/(u t ) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u t ) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which F q [u]/(u t ) is an example) have been a source of many interesting properties . Gulliver and Harada [4] found good examples of ternary codes over F 3 using a particular type of Gray map. Siap and Ray-Chaudhuri in established a relation between codes over F q [u]/(u 2 -a) and codes over F q , which was used to obtained new codes over F 3 and F 5 . In this paper we present a certain generalization of the method used in [4] and , defining a family of metrics for linear codes over F q [u]/(u t ) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F 2 + uF 2 are more powerful than codes obtained by the so-called u-u+v condition. With these tools in hand, we study conditions for self-duality of codes over F q [u]/(u t ). In the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over F q that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over F q starting from a self-dual code over F q [u]/(u t ). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over F q [u]/(u t ) starting from a self-orthogonal code over F q . Our results contain many of the properties studied by Bachoc for self-dual codes over 2. Metric for Codes over F q [u]/(u t ). We will use R(q, t) to denote the commutative ring F q [u]/(u t ). The q t elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the * The project was partially supported by Office of Research of the University of Michigan-Flint.

In this paper, we investigate the structure and properties of duadic, isodual cyclic and formally self-dual codes over the ring R = F_q + vF_q with v2 = v. In addition to the theoretical work on the structure of these codes, we construct examples of good codes over different alphabets from cyclic self-dual and formally self-dual codes over R.

Discrete Mathematics, 2016

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 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.

2018

In this work, extension theorems are generalized to 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) and F42 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 a corresponding result about 3-designs is stated.

In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F_2^m+uF_2^m for m = 1, 2. The duality and distance preserving Gray maps from F4 +uF4 to (F_2 +uF_2)^2 and (F_4)^2 are used to obtain self-dual codes whose binary Gray images are [64,32,12]-extremal self-dual. An F_2+uF_2-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the ?rst examples of codes with gamma=3 and many codes with the rare gamma= 4, 6 parameters are obtained. In addition to these, two hundred ?fty doubly even self dual [96,48,16]-codes with new weight enumerators are obtained from four-circulant codes over F_4 + uF_4. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F_2+uF_2-lifts of binary four circulant codes and a corresponding result about 3-designs is stated.

arXiv (Cornell University), 2015

In this paper, we study the linear codes over the commutative ring R = F q + vF q + v 2 F q and their Gray images, where v 3 = v. We define the Lee weight of the elements of R, we give a Gray map from R n to F 3n q and we give the relation between the dual and the Gray image of a code. This allows us to investigate the structure and properties of self-dual cyclic, formally self-dual and the Gray image of formally self-dual codes over R. Further, we give several constructions of formally self-dual codes over R.

Mathematica Slovaca, 2016

In this work, we study codes over the ring R k,m = F 2 [u, v]/ u k , v m , 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 km 2 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].