Explicit Deep Holes of Reed-Solomon Codes
Jun Zhang, Daqing Wan
TL;DR
This paper investigates deep holes in Reed-Solomon codes, providing new classes and explicit constructions, especially for projective Reed-Solomon codes with specific redundancies and field characteristics.
Contribution
It introduces new classes of deep holes for generalized affine RS codes and explicitly constructs deep holes for PRS codes, including complete characterization for certain redundancies.
Findings
Deep holes for generalized affine RS codes identified under specific conditions.
Explicit constructions of deep holes for PRS codes with redundancy three.
Most deep holes for PRS codes with redundancy four are characterized asymptotically.
Abstract
In this paper, deep holes of Reed-Solomon (RS) codes are studied. A new class of deep holes for generalized affine RS codes is given if the evaluation set satisfies certain combinatorial structure. Three classes of deep holes for projective Reed-Solomon (PRS) codes are constructed explicitly. In particular, deep holes of PRS codes with redundancy three are completely obtained when the characteristic of the finite field is odd. Most (asymptotically of ratio ) of the deep holes of PRS codes with redundancy four are also obtained.
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Taxonomy
TopicsCoding theory and cryptography · Cellular Automata and Applications · graph theory and CDMA systems