New insights solve deep hole problem in reed solomon codes
Deep holes of full-length Reed--Solomon codes
Information Theory
Summary
This paper solves a difficult problem about special 'deep hole' words in Reed-Solomon codes, which are used in error correction. The authors figure out exactly which words qualify as deep holes for most cases, including special cases when the code length is very large. They use advanced math tools like graph theory and algebraic structures to reach their conclusions. Their work also clarifies some other related properties of these codes.
What this means in practice
- •For communications engineers: Determine exact error patterns that are hardest to correct in Reed-Solomon codes for better system design.
- •For coding theorists: Use characterization of deep holes to improve decoding algorithms and bounds in error-correcting code research.
A theory result. No direct application yet.
Authors
Bocong Chen, Jing Huang
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.