Optimal binary codes from $\mathcal{C}_{D}$-codes over a non-chain ring
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| Format: | Preprint |
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2025
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| author | Yadav, Ankit Sarma, Ritumoni Bhagat, Anuj Kumar |
| author_facet | Yadav, Ankit Sarma, Ritumoni Bhagat, Anuj Kumar |
| contents | In \cite{shi2022few-weight}, Shi and Li studied $\mathcal{C}_D$-codes over the ring $\mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes $\mathcal{C}_{D}^{(2)}$ of $\mathcal{C}_{D}$-codes over $\mathcal{R},$ where $D$ is as in \cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of $\mathcal{C}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_09057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal binary codes from $\mathcal{C}_{D}$-codes over a non-chain ring Yadav, Ankit Sarma, Ritumoni Bhagat, Anuj Kumar Information Theory In \cite{shi2022few-weight}, Shi and Li studied $\mathcal{C}_D$-codes over the ring $\mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes $\mathcal{C}_{D}^{(2)}$ of $\mathcal{C}_{D}$-codes over $\mathcal{R},$ where $D$ is as in \cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of $\mathcal{C}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes. |
| title | Optimal binary codes from $\mathcal{C}_{D}$-codes over a non-chain ring |
| topic | Information Theory |
| url | https://arxiv.org/abs/2510.09057 |