Improvements on the Johnson bound for Reed-Solomon Codes

For Reed-Solomon Codes with block length n and dimension k, the Johnson theorem states that for a Hamming ball of radius smaller than n − √ nk, there can be at most O(n2) codewords. It was not known whether for larger radius, the number of code words is polynomial. The best known list decoding algorithm for Reed-Solomon Codes due to Guruswami and Sudan [13] is also known to work in polynomial time only within this radius. In this paper we prove that when k < αn for any constant 0 < α < 1, we can overcome the barrier of the Johnson bound for list-decoding of Reed-Solomon Codes (even if the field size is exponential). More specifically in such a case, we prove that for Hamming ball of radius n − √ nk + c, (for any c> 0) there can be at most O(n c (1 − √ α) 2 +c+2) number of codewords. For any constant c, we describe a polynomial time algorithm to enumerate all of them, thereby also improving on the Guruswami-Sudan’s algorithm. Although the improvement is modest this provides evidence for the first time that the n − √ nk bound is not sacrosanct for such a high rate. We apply our method to obtain sharper bounds on a list recovery problem introduced by Guruswami and Rudra [11] where they establish super polynomial lower bounds on the output size when the list size exceeds ⌈ n k ⌉. We show that even for larger list sizes the problem can be solved in polynomial time for certain values of k. 2 √ α