The complexity of information set decoding

Abstract

Information set decoding is an algorithm for decoding any linear code. Expressions for the complexity of the procedure that are logarithmically exact for virtually all codes are presented. The expressions cover the cases of complete minimum distance decoding, bounded hard-decision decoding, and the important case of bounded soft-decision decoding. It is demonstrated that these results are vastly better than for the trivial algorithms of searching through all codewords or through all syndromes and are significantly better than those for any other general algorithm currently known. For codes over large symbol fields the procedure tends towards a complexity that is subexponential in the symbol size. n-+ct. n F(q , R) = lim-log, E(q , R) where E (q , R) is the number of computations necessary (i.e., we have E (q , R) = q n F (q * R) + o (n)

On the complexity of suboptimal decoding for list and decision feedback schemes

“…Note that the known complexity of optimum decoding is of order 2 min(R,1−R)n . Evseev's result on the decoding complexity was later on improved by several authors [1,4,5].…”

General Methods of Decoding of Linear Codes

“…Papers [3], [5], [6], [12], [13] and [14] introduced new decoding methods and provided theoretical justification of those already known.…”

“…A key result in [13] and [21] states that as the length of the code grows, any k coordinates form an 'almost' information set, i.e., that the codimension of the space of code vectors in any k coordinates is small. This means that the number of code vectors that project identically on any k coordinates, i.e., the size of the list MðÞ for any , is small.…”

“…The following lemma shows that this size grows as q Oðn 1=2 Þ and, therefore, does not contribute to the main term of the complexity estimate. In [7] a slightly better estimate was proved for the corank of square submatrices of a random matrix than the one in [13] and [21]. Lemma 3.2 Let A be a random (k  n) matrix over F q ; k < n; k; n !…”

“…Theorem 3.4 [13], [21]: The covering-set decoding for almost all codes performs minimum distance decoding. The decoding can be implemented by a sequential algorithm with time complexity at most .…”