Research On Linear Codes Over Several Classes Of Finite Non-chain Rings

With the development of error-correcting codes over finite chain rings, studies over some finite non-chain rings also have been considered by some authors. Cyclic codes and constacyclic codes play an important role in error-correcting codes, they have been studied by many authors. In this paper, we mainly study the structure and the properties of several classes of cyclic codes and constacyclic codes over some finite non-chain rings.(1) The (1+u+v) — constacyclic codes over the ring R= F2+uF2+vF2+uvF2 is discussed. Firstly a Gray map of C is defined and is proved that under this map the Gray image of C over R is a binary distance invariant quasi-cyclic code. Also their dual codes are discussed. Then a set of generators of such constacyclic codes for an arbitrary length is determined. Finally, an optimal binary code is obtained from the Gray map.(2) The (l+2s-1u)-constacyclic codes over the ring Z2s,+uZ2s is studied. The structure of such constacyclic codes of odd length is determined. The properties of the Gray map is discussed, it is proved that the image of the (1+2s-1u)— constacyclic codes is a cyclic code over Z2s.The 1 —generator (1+2s-1u) —constacyclic codes is also discussed.(3) we study cyclic codes over the ring Fpm+vFpm+v2Fpm+v3Fpm· We decompose the ring based on the decomposition of chinese remainder theorem. A Gray map over the ring is defined, the relation between the dual and the Gray image of a code is discussed. Finally, we give the structure of cyclic code over such ring.