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In this paper, we use a novel approach to describe generator polynomials of quasi-cyclic (QC) and generalized QC (GQC) codes over finite fields. Our study of QC- and GQC-codes will be general and not only restricted to one-generator codes. We prove that generator polynomials of QC-codes and GQC-codes are unique. Further, we use our results to obtain an expression for the dimensions of QC-codes and GQC-codes. As an application of our construction of these codes, we obtain many optimal linear codes over finite fields GF (2) , GF (3) , GF (4) and GF (5). [ABSTRACT FROM AUTHOR]

In this paper, we introduce double Quadratic Residue Codes (QRC) of length n = p + q for prime numbers p and q in the ambient space F 2 p × F 2 q. We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p = q. We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting. [ABSTRACT FROM AUTHOR]

In this paper, we will study the structure of ℤ 4 R -additive codes where ℤ 4 = { 0 , 1 , 2 , 3 } is the well-known ring of 4 elements and R is the ring given by R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = u , v 2 = v and u v = v u = 0. We will classify all maximum distance separable codes with respect to the Singleton bound (MDSS) over ℤ 4 R. Then we will focus on ℤ 4 R -additive cyclic and constacyclic codes. We will find a unique set of generator polynomials for these codes and determine minimum spanning sets for them. We will also find the generator polynomials for the dual of any ℤ 4 R -additive cyclic or constacyclic code. [ABSTRACT FROM AUTHOR]

Let 𝔽 4 be the field of four elements. We denote by R the commutative ring, with 1 6 elements, 𝔽 4 + v 𝔽 4 : = { a + v b | a , b ∈ 𝔽 4 } with v 2 = v. This work defines linear codes over the ring of mixed alphabets 𝔽 4 R as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over 𝔽 4 R. [ABSTRACT FROM AUTHOR]