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Semidefinite programming bounds for binary codes from a split Terwilliger algebra

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arxiv 2203.06568 v4 pith:X2MHB4NI submitted 2022-03-13 cs.IT math.COmath.IT

Pin-Chieh Tseng , Ching-Yi Lai , Wei-Hsuan Yu This is my paper · ORCID
classification cs.ITmath.COmath.IT
keywords boundssemidefinitealgebraprogrammingterwilligerhammingimproveschrijver
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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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  1. Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials

    math.CO 2026-07 accept novelty 7.0 of 10

    A robust OR polynomial yields Schrijver quasi-tensorization ϑ'(⊠Gi)≤∏ϑ'(Gi)^{C log r log ϑ'(Gi)} and improves Rr(k) to exp(−Ω(k/(r^9(log r)^6))) r^{rk}.

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