Lower Bounds on the Minimum Pseudo-Weight of Linear Codes

Abstract

We discuss two techniques for obtaining lower bounds on the (AWGN channel) pseudo-weight of binary linear codes. Whereas the first bound is based on the largest and second-largest eigenvalues of a matrix associated with the parity-check matrix of a code, the second bound is given by the solution to a linear program. The fundamental polytope/cone [1] turns up in a variety of contexts: when characterizing the valid configurations of graph-covers of factor graphs, when formulating the linear programming decoder, or when looking at the beliefs that are possible for the Bethe free energy associated to a factor graph. It is probably fair to say that one of the most important parameters that characterize the fundamental polytope/cone is the minimum pseudo-weight of all the pseudo-codewords that lie in the fundamental polytope/cone. The AWGNC pseudo-weight [2, 1] of a pseudo-codeword x is defined to be wp(x) 4 = w p (x) 4 = ||x|| 1 ||x|| 2 . In the following, w p (H) will denote the minimum AWGNC pseudo-weight of a linear code C defined by the parity-check matrix H and w H (C) will denote the minimum Hamming weight of a linear code C. (Given a code, note that the minimum Hamming weight is a function of the code whereas the minimum pseudo-weight is a function of the parity-check matrix describing the code.) In [1] we discussed two ways of obtaining upper bounds on the minimum AWGNC pseudo-weight: one of them was based on searching for low-weight pseudo-codewords in the fundamental cone, the other was based on the so-called canonical completion. In this paper we now introduce two techniques for obtaining lower bounds. The first one (Th. 1) is a purely algebraic eigenvalue-based bound that turns out to have the same form as the bit-oriented lower bound given by Tanner [3] for the minimum Hamming weight of a binary code. (Therefore, parity-check matrices that lead there to a non-trivial bound give also here a nontrivial bound.) The second bound (Claim 3) is a linear-programming-based bound which was originally very much inspired by the linear programming lower bound on the minimum Hamming weight as presented by Tanner [3]. But finally, its form is quite different. Actually, the present form of the linear program reminds much more of the “lift and project” technique in [4, Sec. 5.4.2] which was used to obtain a modification of the linear programming decoder. But the approach in [4] is used to constrain the fundamental polytope whereas we are interested in relaxing the fundamental polytope. Note moreover that in [3] and in [4] an important ingredient is the relation xi = x 2 i (which holds because the components of the vector x were desired to be 0 or 1), but this does not hold anymore for components of 1Both authors were supported by NSF Grants CCR 99-84515 and CCR 01-05719. pseudo-codewords. Theorem 1 Let C be a (j, k)-regular code of length n defined by the parity-check matrix H and let the corresponding Tanner graph have one component. Let L 4 = HH and let μ1 and μ2 be the largest and second-largest eigenvalue, respectively, of L. Then the minimum Hamming weight and the minimum AWGNC pseudo-weight are lower bounded by w min H (C) ≥ w min p (H) ≥ n · 2j − μ2 μ1 − μ2 . Corollary 2 Consider a binary code of length n whose automorphism group is two-transitive on the bits and whose dual code has minimum Hamming weight w H ⊥ (C). Let H be the matrix consisting of all vectors in the dual code whose Hamming weight equals w H ⊥ (C). Then, w min H (C) ≥ w min p (H) ≥ n − 1

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@inproceedings{Vontobel2004LowerBO, title={Lower Bounds on the Minimum Pseudo-Weight of Linear Codes}, author={Pascal O. Vontobel and Ralf Koetter}, year={2004} }