Some Linear Codes Over Finite Rings

Codes over finite rings have been studied extensively during the last three decades.Negacyclic repeated-root codes over GR(2a,m) have been characterized:R2(a,m,-1)= (?) is a chain ring with maximal ideal(x + 1).Furthermore,these results have been generalized to A-constacyclic codes for any unit ? of the form 4z-1,z ? GR(2a,m).The notion of trace codes with a defining set has been extended from finite fields to finite rings,and many linear trace codes over finite fields and rings have been produced by selecting different defining sets.This thesis is devoted to the study of the minimum Hamming distances and the weight distributions of different,types of linear codes(?-constacyclic codes,trace codes,repeated-root codes)over finite rings.In Chapter 3,we study the structure of the ring Rp(a,m,?)=(?),where? is a unit of GR(pa,m),and we give a necessary and sufficient condition for the ring Rp(a,m,?)to be a chain ring.Also,we study the structures of ?-constacyclic codes of length ps over the Galois ring GR(pa,m)for all the cases where Rp(a,m,?)is a chain ring.Furthermore,we provide the necessary and sufficient conditions for the existence of self orthogonal and self dual ?-constacyclic codes.Among others,for any prime p,the structure of Rp(a,m,?) is used to establish the Hamming and homogeneous distances of ?-constacyclic codes.In Chapter 4,we define the trace codes C(q,e,m,m1) over the ring R= Fq + uFq with the defining set L = Co(e,qm)+ Fqm1,where e is a divisor of g-1 and CO(e,qm)is the cyclotomic class of order e.Also,we show that the Lee weight distribution of these codes are related to the gcd(e,m),and determine it for a small value of gcd(e,m).Moreover,when gcd(e,m)= 1 the Gray images of the codes attain the Griesmer bound,and in some few cases their images are optimal codes for a given length and dimension.When gcd(e,m)= 2,3 or 4 we construct new families of trace codes with at most five weights.In Chapter 5.we give all the minimum Hamming distances of cyclic codes of length ps over the ring R = Fpm + uFpm,where u2 = 0.Besides,we establish an isometry between cyclic codes and a-constacyclic codes over R where ?? Fp.*As a consequence of this,we determine the minimum Hamming distances of all a-constacyclic codes of length ps over R.