Mathematical proof note · Version 1.0 · October 8, 2026
Exact Half-Rate Points of the q-Ary Delsarte Bound
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
We determine the inverse half-rate distance of the complete Delsarte linear programming bound over every fixed integer alphabet. The same distance is the exact half-rate point of the optimization over all finite positive auxiliary matrices. Weighted Krawtchouk concentration gives the lower bound, with intersecting circles, tangent circles, and an annulus determining the nonbinary constants. A representation-theoretic transfer and finite positive-tensor constructions give the matching upper bound: Gram duality converts the self-dual Potts energy into auxiliary inputs with normalized information tending to one half.
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Keywords
q-ary codes; Delsarte linear programming bound; half-rate distance; Krawtchouk polynomials; auxiliary matrices; Potts model.
中文关键词：q元码；Delsarte线性规划界；半码率距离；Krawtchouk多项式；辅助矩阵；Potts模型。
Suggested citation
S. Yang, "Exact Half-Rate Points of the q-Ary Delsarte Bound," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/qary-delsarte-half-rate/proof.pdf