25 research outputs found
The hardness of decoding linear codes with preprocessing
The problem of maximum-likelihood decoding of linear block codes is known to be hard. The fact that the problem remains hard even if the code is known in advance, and can be preprocessed for as long as desired in order to device a decoding algorithm, is shown. The hardness is based on the fact that existence of a polynomial-time algorithm implies that the polynomial hierarchy collapses. Thus, some linear block codes probably do not have an efficient decoder. The proof is based on results in complexity theory that relate uniform and nonuniform complexity classes
Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard
Maximum-likelihood decoding is one of the central algorithmic problems in
coding theory. It has been known for over 25 years that maximum-likelihood
decoding of general linear codes is NP-hard. Nevertheless, it was so far
unknown whether maximum- likelihood decoding remains hard for any specific
family of codes with nontrivial algebraic structure. In this paper, we prove
that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon
codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes
remains hard even with unlimited preprocessing, thereby strengthening a result
of Bruck and Naor.Comment: 16 pages, no figure
Computing coset leaders and leader codewords of binary codes
In this paper we use the Gr\"obner representation of a binary linear code
to give efficient algorithms for computing the whole set of coset
leaders, denoted by and the set of leader codewords,
denoted by . The first algorithm could be adapted to
provide not only the Newton and the covering radius of but also to
determine the coset leader weight distribution. Moreover, providing the set of
leader codewords we have a test-set for decoding by a gradient-like decoding
algorithm. Another contribution of this article is the relation stablished
between zero neighbours and leader codewords