Small codes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914709982150656 |
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| author | Balla, Igor |
| author_facet | Balla, Igor |
| contents | Determining the maximum number of unit vectors in $\mathbb{R}^r$ with no pairwise inner product exceeding $α$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all $α\leq 0$ and in this paper, we show that the maximum is $(2+o(1))r$ for all $0 \leq α\ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the exponent $-2/3$ is best possible. As a consequence, we conclude that when $j \ll r^{1/3}$, a $q$-ary code with block length $r$ and distance $(1-1/q)r - j$ has size at most $(2 + o(1))(q-1)r$, which is tight up to the multiplicative factor $2(1 - 1/q) + o(1)$ for any prime power $q$ and infinitely many $r$. When $q = 2$, this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent $1/3$ is best possible. Finally, using a recently discovered connection to $q$-ary codes, we obtain analogous results for set-coloring Ramsey numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19047 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small codes Balla, Igor Combinatorics Information Theory Metric Geometry 52C17 (Primary) 94B65, 05B40, 05D10 (Secondary) E.4 Determining the maximum number of unit vectors in $\mathbb{R}^r$ with no pairwise inner product exceeding $α$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all $α\leq 0$ and in this paper, we show that the maximum is $(2+o(1))r$ for all $0 \leq α\ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the exponent $-2/3$ is best possible. As a consequence, we conclude that when $j \ll r^{1/3}$, a $q$-ary code with block length $r$ and distance $(1-1/q)r - j$ has size at most $(2 + o(1))(q-1)r$, which is tight up to the multiplicative factor $2(1 - 1/q) + o(1)$ for any prime power $q$ and infinitely many $r$. When $q = 2$, this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent $1/3$ is best possible. Finally, using a recently discovered connection to $q$-ary codes, we obtain analogous results for set-coloring Ramsey numbers. |
| title | Small codes |
| topic | Combinatorics Information Theory Metric Geometry 52C17 (Primary) 94B65, 05B40, 05D10 (Secondary) E.4 |
| url | https://arxiv.org/abs/2305.19047 |