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新论文 · v1arXiv:2611.13481

Euclidean Hulls of Double Cyclic Codes of Arbitrary Block Lengths

Awadhesh Kumar Shukla, Om Prakash Pandey, Sachin Pathak, Ashish Upadhyay

原文摘要

Gao et al. \cite{GWF23} recently explored the Euclidean hulls of double cyclic codes over $\mathbb{Z}_2$. While they found answers for non-separable codes with coprime lengths ($\gcd(r, s) = 1$), the non-coprime case ($\gcd(r, s) > 1$) remained an open question. This paper completely solves that problem for all possible block lengths. We calculate the exact generator polynomials for the hull and show the precise conditions needed for a non-separable code to turn into a separable one. Using this, we show how to construct non-separable LCD codes. We also include several numerical examples to confirm our findings.

cs.IT

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版本更新 · v2arXiv:2404.00613

On $(θ, Θ)$-cyclic codes and their applications in constructing QECCs

Awadhesh Kumar Shukla, Sachin Pathak, Om Prakash Pandey, Vipul Mishra, Ashish Kumar Upadhyay

原文摘要

Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with $u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of $(θ, Θ)$-cyclic codes of block length $(r,s )$ over $\mathbb{F}_qR.$ Specifically, we analyze the structure of these codes as left $R[x:Θ]$-submodules of $\mathfrak{R}_{r,s} = \frac{\mathbb{F}_q[x:θ]}{\langle x^r-1\rangle} \times \frac{R[x:Θ]}{\langle x^s-1\rangle}$. Our investigation involves determining generator polynomials and minimal generating sets for this family of codes. Further, we discuss the algebraic structure of separable codes. A relationship between the generator polynomials of $(θ, Θ)$-cyclic codes over $\mathbb F_qR$ and their duals is established. Moreover, we calculate the generator polynomials of dual of $(θ, Θ)$-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from $(θ, Θ)$-cyclic codes of block length $(r,s)$ over $\mathbb{F}_qR$. We support our theoretical results with illustrative examples.

cs.IT

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