Deep holes and MDS extensions of Reed-Solomon codes
Krishna Kaipa
TL;DR
This paper investigates deep holes in Reed-Solomon codes, establishing an equivalence with MDS extensions, leading to new classifications of deep holes for codes with specific dimensions and redundancies.
Contribution
It introduces an equivalence between deep holes and MDS extensions, enabling comprehensive classification results for certain Reed-Solomon codes.
Findings
Classified deep holes for codes with dimension > half the alphabet size.
Provided a complete classification for codes with redundancy three.
Improved upon previous results in the classification of deep holes.
Abstract
We study the problem of classifying deep holes of Reed-Solomon codes. We show that this problem is equivalent to the problem of classifying MDS extensions of Reed-Solomon codes by one digit. This equivalence allows us to improve recent results on the former problem. In particular, we classify deep holes of Reed-Solomon codes of dimension greater than half the alphabet size. We also give a complete classification of deep holes of Reed Solomon codes with redundancy three in all dimensions.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research