On the codes over the mix alphabet

This thesis consists of six chapters. In the first chapter, a literature review on coding theory and some specific linear codes are discussed. In the second chapter, fundamental definitions and theorems related to algebraic concepts, coding theory and skew constacyclic codes defined over finite field and certain families of rings are given. In the third chapter, a new mixed alphabet Δ=F_q B is constructed, where F_q is a finite field and B=F_q [v]/〈v^3-v〉 is a family of finite rings. The algebraic structure of skew constacyclic codes over Δ is specifically studied. Additionally, the concept of separable codes is introduced, and the duals and Gray images of skew constacyclic codes are obtained. In the fourth chapter, a new mixed alphabet F_q R_1 R_2 is constructed using finite field F_q and two families of finite rings R_1=F_q+u_1 F_q, u_1^2=β_1 u_1 and R_2=F_q+u_1 F_q+u_2 F_q, u_1^2=β_1 u_1, u_2^2=β_2 u_2, u_1 u_2=u_2 u_1=0, β_1,β_2∈F_q^*. Linear codes defined over F_q R_1 R_2 are investigated. In particular, the algebraic structure of skew constacyclic codes over F_q R_1 R_2, the concept of separable codes, their duals and Gray images are studied. In the fifth chapter, a new family of finite rings is constructed as S_(q,k)=F_q [w_1,w_2,…,w_k ]/〈w_j^2=w_j,w_i w_j=w_j w_i,w_1 w_2-w_2 w_1 〉 for i=3,…,k, j=1,2,…,k, i≠j k≥3, F_q is a finite field. Skew constacyclic codes over S_(q,k) are studied for k≥3. Furthermore, a new mixed alphabet F_q S_(q,k) is constructed, and linear codes over F_q S_(q,k) are investigated. Specifically, the algebraic structure of skew constacyclic codes, the concept of separable codes, their duals and Gray images are explored. The final chapter includes the conclusion and discussions.