On binary codes with distances $d$ and $d+2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917594844364800 |
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| author | Landjev, Ivan Vorob'ev, Konstantin |
| author_facet | Landjev, Ivan Vorob'ev, Konstantin |
| contents | We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size $\sim\frac{n^2}{\frac{d}{2}(\frac{d}{2}+1)}$ can be obtained from $2$-packings of an $n$-element set by blocks of cardinality $\frac{d}{2}+1$. This value is far from the upper bound $A_2(n,\{d_1,d_2\})\le1+{n\choose2}$ proved recently by Barg et al.
In this paper we prove that for every fixed $d$ ($d$ even) there exists an integer $N(d)$ such that for every $n\ge N(d)$ it holds $A_2(n,\{d,d+2\})=D(n,\frac{d}{2}+1,2)$, or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on $N(d)$ for $d=4$ and $d=6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_13420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On binary codes with distances $d$ and $d+2$ Landjev, Ivan Vorob'ev, Konstantin Combinatorics 05A05, 05A20, 94B25, 94B65 We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size $\sim\frac{n^2}{\frac{d}{2}(\frac{d}{2}+1)}$ can be obtained from $2$-packings of an $n$-element set by blocks of cardinality $\frac{d}{2}+1$. This value is far from the upper bound $A_2(n,\{d_1,d_2\})\le1+{n\choose2}$ proved recently by Barg et al. In this paper we prove that for every fixed $d$ ($d$ even) there exists an integer $N(d)$ such that for every $n\ge N(d)$ it holds $A_2(n,\{d,d+2\})=D(n,\frac{d}{2}+1,2)$, or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on $N(d)$ for $d=4$ and $d=6$. |
| title | On binary codes with distances $d$ and $d+2$ |
| topic | Combinatorics 05A05, 05A20, 94B25, 94B65 |
| url | https://arxiv.org/abs/2402.13420 |