New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code
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arXiv
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| Format: | Preprint |
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2025
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| author | Davydov, Alexander A. Marcugini, Stefano Pambianco, Fernanda |
| author_facet | Davydov, Alexander A. Marcugini, Stefano Pambianco, Fernanda |
| contents | We present five new infinite families of linear near-MDS codes uniformly packed in the wide sense (UPWS). These codes are also almost perfect multiple coverings of the deep holes or farthest-off points (APMCF), i.e.\ the vectors lying at distance $R$ (covering radius) from the code. The families are constructed by $m$-lifting when one takes a starting code $C$ over the ground Galois field $\F_q$ with a parity check matrix $H(C)$ and then considers the codes $C_m$ over $F_{q^m}$, $m\ge2$, with the same parity check matrix $H(C)$. As starting codes we used the ternary perfect Golay code and codes obtained by its extension and puncturing. To prove the needed combinatorial properties (UPWS and APMCF), we used the $m$-lifting of the dual codes and features of near-MDS codes. A general theorem on infinite families of UPWS near-MDS codes is proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code Davydov, Alexander A. Marcugini, Stefano Pambianco, Fernanda Combinatorics 94B25, 94B60, 94B05 We present five new infinite families of linear near-MDS codes uniformly packed in the wide sense (UPWS). These codes are also almost perfect multiple coverings of the deep holes or farthest-off points (APMCF), i.e.\ the vectors lying at distance $R$ (covering radius) from the code. The families are constructed by $m$-lifting when one takes a starting code $C$ over the ground Galois field $\F_q$ with a parity check matrix $H(C)$ and then considers the codes $C_m$ over $F_{q^m}$, $m\ge2$, with the same parity check matrix $H(C)$. As starting codes we used the ternary perfect Golay code and codes obtained by its extension and puncturing. To prove the needed combinatorial properties (UPWS and APMCF), we used the $m$-lifting of the dual codes and features of near-MDS codes. A general theorem on infinite families of UPWS near-MDS codes is proved. |
| title | New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code |
| topic | Combinatorics 94B25, 94B60, 94B05 |
| url | https://arxiv.org/abs/2502.10223 |