Complexity of Decoding Positive-Rate Reed-Solomon Codes

Abstract. The complexity of maximum likelihood decoding of the ReedSolomon codes [q -1, k]q is a well known open problem. The only known result [4] in this direction states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, we remove the rate restriction and prove that the same complexity result holds for any positive information rate. In particular, this resolves an open problem left in [4], and rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of Reed-Solomon codes of any rate under a well known cryptographical hardness assumption. As a side result, we give an explicit construction of Hamming balls of radius bounded away from the minimum distance, which contain exponentially many codewords for Reed-Solomon code of any positive rate less than one. The previous constructions in [2][7] only apply to Reed-Solomon codes of diminishing rates. We also give an...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Qi Cheng, Daqing Wan
• Keywords: CORR 2008, Education, Information Rate, Open Problem, Reed-Solomon Code