Deep holes of full-length Reed--Solomon codes

Bocong Chen; Jing Huang

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arXiv · 2610.29429

Deep holes of full-length Reed--Solomon codes

Bocong Chen·Jing Huang

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Abstract

For every prime power $q>19683$, we settle the Cheng--Murray deep-hole problem for full-length Reed--Solomon codes in every dimension $2\le k\le q-2$, with the characteristic-two cases stated explicitly. For odd $q$, the deep holes are precisely the words of degree $k$. For even $q$ and $k\ge3$, the same conclusion holds except for an additional family when $k=q-3$. In even characteristic and dimension two, every deep hole has a unique expression $a h(X)+bX+c$, where $a\ne0$ and $h$ is a normalized $o$-polynomial. The proof combines quadratic quotient algebras, graph-theoretic weighted independent-set inequalities, and planar arc completion. We also obtain normal rational curve rigidity in the corresponding nonexceptional dimensions, the projective Reed--Solomon covering radii, and distinct-coordinate solvability and remainder criteria.

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BibTeX RIS

Bocong Chen, Jing Huang. 2026-10-21. Deep holes of full-length Reed--Solomon codes. https://arxiv.org/abs/2610.29429

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