A deterministic reduction for the gap minimum distance problem: [extended abstract]
Proceedings of the forty-first annual ACM symposium on Theory of computing
A simple deterministic reduction for the gap minimum distance of code problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Generalised jacobians in cryptography and coding theory
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
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The complexity of maximum likelihood decoding of theReed-Solomon codes [q - 1,k]q is a well known open problem. Theonly known result [4] in this direction states that it is at leastas hard as the discrete logarithm in some cases where theinformation rate unfortunately goes to zero. In this paper, weremove the rate restriction and prove that the same complexityresult holds for any positive information rate. In particular, thisresolves an open problem left in [4], and rules out the possibilityof a polynomial time algorithm for maximum likelihood decodingproblem of Reed-Solomon codes of any rate under a well knowncryptographical hardness assumption. As a side result, we give anexplicit construction of Hamming balls of radius bounded away fromthe minimum distance, which contain exponentially many codewordsfor Reed-Solomon code of any positive rate less than one. Theprevious constructions in [2][7] only apply to Reed-Solomon codesof diminishing rates. We also give an explicit construction ofHamming balls of relative radius less than 1 which containsubexponentially many codewords for Reed-Solomon code of rateapproaching one.