Codes over a Non Chain Ring with Some Applications

An significant milestone study in coding theory recognized to be the paper written by Hammons at al. [1]. Fields are useful area for constructing codes but after the study [1] finite ring have received a great deal of attention. Most of the studies are concentrated on the case with codes over finite chain rings. However, optimal codes over nonchain rings exist (e.g see [2].) In [3], et al. studied the algebraic structure of cyclic codes over F2 + vF2, where v 2 = v. Zhu and Wang studied a class of constacyclic codes in [4]. Recently, D.Boucher et al. thought over non commutative rings. They examined the algebraic structure of ideals of ring as cyclic codes and they found codes which has linear codes under Gray map. The difficulty on working in non commutative case is that the polynomial ring is no longer left or right. In this work, we classified all constacyclic codes over F4 + vF4 and gave the relation between constacyclic codes and Gray images. Constacyclic codes are generated principally. Also we deal with the constacyclic codes approach with non-commutative case. Therefore we sorted out skew constacyclic codes. The structure of all skew constacyclic codes is completely determined. Shiromoto and Storme gave a Griesmer type bound for linear codes over finite quasi-Frobenius rings [5]. Since R is finite quasi-Frobenius ring, we obtain some DNA codes over R that attain the Griesmer bound over R. We have written some algorithms and programs in MAGMA package and Mathematica software to construct these codes.