Mathematical proof note · Version 1.0 · October 8, 2026
The Covering-Radius Formula for Projective Reed-Solomon Codes
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
The covering radius of a projective Reed–Solomon code is governed by whether its dual normal rational curve admits one further point. We prove the radius formula for every prime power q ≥ 4 and every dimension 2 ≤ k ≤ q-2: the radius is q-k, except that it is q+1-k for even q and k ∈ {2,q-2}. The proof combines a finite-field complement identity with a selected root replacement. Two degree-bounded interpolation polynomials agree on too many nodes, while their highest coefficients differ. A projective involution treats the complementary dimension range. The argument includes both exceptional endpoints and gives the complete proof of the stated radius formula.
Full text and source
Keywords
projective Reed-Solomon codes; covering radius; MDS codes; normal rational curves; finite geometry; deep holes.
中文关键词：射影里德所罗门码；覆盖半径；最大距离可分码；正规有理曲线；有限几何；深洞。
Suggested citation
S. Yang, "The Covering-Radius Formula for Projective Reed-Solomon Codes," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/projective-reed-solomon-covering-radius/proof.pdf