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In this paper, we study the class of ℤp-double cyclic codes of length n = r + s. We give a closed formula for the number of ℤp -double cyclic codes of length n = r + s, for any integers r and s that are relatively prime to p. Moreover, we give a closed formula for the number of quasi-cyclic (QC) codes of length n = 2s and index 2. We also provide formulas for the number of separable and non-separable ℤp-double cyclic codes of length n. In order to illustrate the results, we calculate the number of some codes with different r and s. Moreover, we list optimal parameter ℤ2-double cyclic codes for specific values of r and s. [ABSTRACT FROM AUTHOR]

This paper considers a new alphabet set, which is a ring that we call 4 R , to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize 4 R -skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over 4 are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of R -skew cyclic codes which are reversible complement. [ABSTRACT FROM AUTHOR]