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Semidefinite programming bounds for binary codes from a split Terwilliger algebra

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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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Authors and Affiliations

  1. Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, 30010, Taiwan

    Pin-Chieh Tseng & Ching-Yi Lai

  2. Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 30010, Taiwan

    Pin-Chieh Tseng

  3. Department of Mathematics, National Central University, Taoyüan, 32001, Taiwan

    Wei-Hsuan Yu

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  1. Pin-Chieh Tseng
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Correspondence to Pin-Chieh Tseng.

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Communicated by G. Korchmaros.

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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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