Abstract
We study the upper bounds for A(n, d), the maximum size of codewords with length n and Hamming distance at least d. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound A(n, d). We derive more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver’s semidefinite programming bounds on A(n, d). In particular, we improve the semidefinite programming bounds on A(18, 4) to 6551.
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Acknowledgements
We would like to thank Dion Gijswijt, Alexander Schrijver, and Hajime Tanaka for helpful discussions. We would also like to thank the anonymous referees for their valuable comments. PCT and CYL were supported by the Ministry of Science and Technology (MOST) in Taiwan under Grant MOST110-2628-E-A49-007. WHY was supported by MOST under Grant109-2628-M-008-002-MY4.
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Tseng, PC., Lai, CY. & Yu, WH. Semidefinite programming bounds for binary codes from a split Terwilliger algebra. Des. Codes Cryptogr. 91, 3241–3262 (2023). https://doi.org/10.1007/s10623-023-01250-4
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DOI: https://doi.org/10.1007/s10623-023-01250-4