ABSTRACT. 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 maximum-likelihood (ML) decoding of a binary linear code over a discrete memoryless channel as a linear programming problem. We translate matroid-theoretic results of Grötschel and Truemper from the combinatorial optimization literature to give examples of non-trivial 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 Koetter-Vontobel fundamental polytope derived from the entire dual code C ⊥. However, we also show that such families of codes are not good in a coding-theoretic sense — either their dimension or their minimum distance must grow sub-linearly with codelength. As a consequence, we have that decoding by linear programming, when applied to good codes, cannot avoid failing occasionally due to the presence of pseudocodewords. 1.

Abstract. We relate the notion of matroid pathwidth to the minimum trellis state-complexity (which we term trellis-width) of a linear code, and to the pathwidth of a graph. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of determining the pathwidth of a representable matroid is NP-hard. Consequently, the problem of computing the trellis-width of a linear code is also NP-hard. For a finite field F, we also consider the class of F-representable matroids of pathwidth at most w, and correspondingly, the family of linear codes over F with trellis-width at most w. These are easily seen to be minor-closed. Since these matroids (and codes) have branchwidth at most w, a result of Geelen and Whittle shows that such matroids (and the corresponding 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. Key words. Matroids, pathwidth, linear codes, trellis complexity, NP-hard. AMS subject classifications. 05B35, 94B05 1. Introduction. The

In the first part of this paper, the basic theory of interleavers is revisited in a semi-tutorial manner, and extended to encompass noncausal interleavers. The parameters that characterize the interleaver behavior (like delay, latency, and period) are clearly defined. The input--output interleaver code is introduced and its complexity studied. Connections among various interleaver parameters are explored. The classes of convolutional and block interleavers are considered, and their practical implementation discussed. In the second part, the trellis complexity of turbo codes is tied to the complexity of the constituent interleaver. A procedure of complexity reduction by coordinate permutation is presented, together with some examples of its application.

. Generalized Hamming weight hierarchies and permutation-optimal trellis decoders are found for several extremal self-dual 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 self-dual codes in general, and is particularly straightforward when certain subcodes meet the Griesmer bound with equality. These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32; 16;8] binary self-dual 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, Conway-Pless codes, double chain condition, generalized Hamming weights, unique codes. 1. Introduction Representations of block codes by trellises allow comput...

Ordered binary decision diagrams (OBDDs) are graph-based data structures for representing Boolean functions. They have found widespread use in computer-aided design and in formal verification of digital circuits. Minimal trellises are graphical representations of error-correcting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphical models, as follows. Let C be a binary code of length n, and let f C (x 1 ; : : : ; x n ) be the Boolean function that takes the value 0 at x 1 ; : : : ; x n if and only if (x 1 ; : : : ; x n ) 2 C . Given this natural oneto -one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code C with minimum distance d ? 1 is isomorphic to the single-terminal OBDD for its Boolean indicator function f C (x 1 ; : : : ; x n ). Prior to this result, the extensive research during the past decade on binary decision diagrams -- in computer engineering -...

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 NP-hard 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).

We present a method for maximum likelihood decoding of the (48; 24; 12) quadratic residue code. This method is based on projecting the code onto a subcode with an acyclic Tanner graph, and representing the set of coset leaders by a trellis diagram. This results in a two level coset decoding which can be considered a systematic generalization of the Wagner rule. We show that unlike the (24; 12; 8) Golay code, the (48; 24; 12) code does not have a Pless-construction which has been an open question in the literature. It is determined that the highest minimum distance of a (48; 24) binary code having a Pless-construction is 10, and up to equivalence there are three such codes.

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 maximum-rate 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.