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Small codes.
- Source :
-
Bulletin of the London Mathematical Society . May2024, Vol. 56 Issue 5, p1680-1686. 7p. - Publication Year :
- 2024
-
Abstract
- Determining the maximum number of unit vectors in Rr$\mathbb {R}^r$ with no pairwise inner product exceeding α$\alpha$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all α⩽0$\alpha \leqslant 0$, and in this paper, we show that the maximum is (2+o(1))r$(2+o(1))r$ for all 0⩽α≪r−2/3$0 \leqslant \alpha \ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the exponent −2/3$-2/3$ is best possible. As a consequence, we obtain an upper bound on the size of a q$q$‐ary code with block length r$r$ and distance (1−1/q)r−o(r1/3)$(1-1/q)r - o(r^{1/3})$, which is tight up to the multiplicative factor 2(1−1/q)+o(1)$2(1 - 1/q) + o(1)$ for any prime power q$q$ and infinitely many r$r$. When q=2$q = 2$, this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent 1/3$1/3$ is best possible. Finally, using a recently discovered connection to q$q$‐ary codes, we obtain analogous results for set‐coloring Ramsey numbers. [ABSTRACT FROM AUTHOR]
- Subjects :
- *CODING theory
*BLOCK codes
*PROTHROMBIN
*GEOMETRY
*EXPONENTS
Subjects
Details
- Language :
- English
- ISSN :
- 00246093
- Volume :
- 56
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Bulletin of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 177040736
- Full Text :
- https://doi.org/10.1112/blms.13019