Some New Optimal Skew Cyclic Codes With Derivation

Abstract

Our study included a class of cyclic codes named $\delta _{\alpha,\zeta }-$ cyclic codes over the ring $\mathcal {R}=\mathbb {F}_{2^{m}} + u\mathbb {F}_{2^{m}}+u^{2}\mathbb {F}_{2^{m}}$ , where m is an odd positive integer with $u^{3}=1$ . These codes are modules over the ring $\mathcal {R}{[x,\alpha,\delta _{\alpha,\zeta }]}$ with automorphism $\alpha $ and $\alpha $ -derivation $\delta _{\alpha,\zeta }$ (where $\zeta \in \mathcal {R}$ ). The structure of ring $\mathcal {R}{[x,\alpha,\delta _{\alpha,\zeta }]}$ is presented to develop linear codes over the ring. The center of the ring is characterized for all inner $\alpha $ -derivations $\delta _{\alpha,\zeta }$ where $\zeta $ is taken as an arbitrary element of the ring fixed by $\alpha $ . In addition, an algorithm was developed and implemented in MAGMA to perform division in this ring. The generator and parity-check matrices for $\delta _{\alpha,\zeta }-$ cyclic codes are proposed, and examples of new optimal $\delta _{\alpha,\zeta }-$ cyclic codes with better minimum Hamming distance than those already existing in the Markus Grassl code tables are provided.

Keywords

• Cyclic code
• derivation
• generator matrix
• right divisor
• singleton bound
• skew cyclic codes