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13
Algorithmic Complexity in Coding Theory and the Minimum Distance Problem
, 1997
"... We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van T ..."
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Cited by 44 (2 self)
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We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van Tilborg, dating back to 1978. Extensions and applications of this result to other problems in coding theory are discussed.
A decomposition theory for binary linear codes
, 2008
"... The decomposition theory of matroids initiated by Paul Seymour in the 1980’s has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for binary linear codes. In this paper, we give an overview of ..."
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Cited by 17 (3 self)
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The decomposition theory of matroids initiated by Paul Seymour in the 1980’s has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for binary linear codes. In this paper, we give an overview of this code decomposition theory, and discuss some of its implications in the context of the recently discovered formulation of maximumlikelihood (ML) decoding of a binary linear code over a discrete memoryless channel as a linear programming problem. We translate matroidtheoretic results of Grötschel and Truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ML decoding problem can be solved in time polynomial in the length of the code. One such family is that consisting of codes C for which the codeword polytope is identical to the KoetterVontobel fundamental polytope derived from the entire dual code C ⊥. However, we also show that such families of codes are not good in a codingtheoretic sense — either their dimension or their minimum distance must grow sublinearly with codelength.
Interleaver properties and their applications to the trellis complexity analysis of turbo codes
, 2001
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The "Art of Trellis Decoding" is Computationally Hard  for Large Fields
 IEEE TRANS. INFORM. THEORY
, 1998
"... The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NPhard for all three ..."
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Cited by 3 (0 self)
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The problem of minimizing the trellis complexity of a code by coordinate permutation is studied. Three measures of trellis complexity are considered: the total number of states, the total number of edges, and the maximum state complexity of the trellis. The problem is proven NPhard for all three measures, provided the field over which the code is specified is not fixed. We leave open the problem of dealing with the case of a fixed field, in particular GF 2).
Trellis Structure and Higher Weights of Extremal SelfDual Codes
 Des., Codes, Cryptogr
, 1999
"... . Generalized Hamming weight hierarchies and permutationoptimal trellis decoders are found for several extremal selfdual codes. The latter problem involves finding chains of subcodes that allow construction of a uniformly efficient permutation. The task of finding such chains of subcodes is shown ..."
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. Generalized Hamming weight hierarchies and permutationoptimal trellis decoders are found for several extremal selfdual codes. The latter problem involves finding chains of subcodes that allow construction of a uniformly efficient permutation. The task of finding such chains of subcodes is shown to be substantially simplifiable in the case of selfdual codes in general, and is particularly straightforward when certain subcodes meet the Griesmer bound with equality. These results are used to characterize the permutationoptimal trellises and generalized Hamming weights for all [32; 16;8] binary selfdual codes and for several other codes. The number of uniformly efficient permutations for the [24; 12;8]Golay code, and a lower bound on the number for the [48; 24;12] quadratic residue code, are found. Keywords: Chain condition, ConwayPless codes, double chain condition, generalized Hamming weights, unique codes. 1. Introduction Representations of block codes by trellises allow comput...
Links Between Complexity Theory and Constrained Block Coding
"... The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. The complexities of several fundamental problems in constrained block coding are shown to be complete in various classes of the existing complexity theoretic ..."
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The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. The complexities of several fundamental problems in constrained block coding are shown to be complete in various classes of the existing complexity theoretic structure. The results include (relatively rare) , Eva , and NPm'completeness results. Two t3'pes of prob lems are considered: (1) the problem of designing encoder and decoder circuits using minimum or approximately minimum hardware for a given constraint and a given rate; (2) computing the maximum rate of a block code for a given constraint and codeword length. In both cases, a constraint is specified by a deterministic finite state transition dia gram. Another question studied is whether maximumrate block codes can always be implemented by encoders and decoders of polynomial size. The answer to this question is shown to be closely' related to the complexit3, of PP.
The "Art of Trellis Decoding" is NPHard ⋆
"... Abstract. Given a linear code C, the fundamental problem of trellis decoding is to find a coordinate permutation of C that yields a code C ′ whose minimal trellis has the least statecomplexity among all codes obtainable by permuting the coordinates of C. By reducing from the problem of computing t ..."
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Abstract. Given a linear code C, the fundamental problem of trellis decoding is to find a coordinate permutation of C that yields a code C ′ whose minimal trellis has the least statecomplexity among all codes obtainable by permuting the coordinates of C. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of finding such a coordinate permutation is NPhard, thus settling a longstanding conjecture.
On Codes of Bounded Trellis Complexity *
"... AbstractIn this paper, we initiate a structure theory of linear codes with bounded trellis complexity. The theory is based on the observation that the family of linear codes over Fq, some permutation of which has trellis statecomplexity at most w, is a minorclosed family. It then follows from a ..."
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AbstractIn this paper, we initiate a structure theory of linear codes with bounded trellis complexity. The theory is based on the observation that the family of linear codes over Fq, some permutation of which has trellis statecomplexity at most w, is a minorclosed family. It then follows from a deep result of matroid theory that such codes are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. We also give a polynomialtime algorithm for determining whether or nor a given code has a permutation with statecomplexity at most 1.