On Some Polynomials Related to Weight Enumerators of Linear Codes

Abstract

Abstract.A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model. Key words. all-terminal reliability, Greene's theorem, linear code, linear matroid, Potts model, rank polynomial, Tutte polynomial, weight enumerator AMS subject classifications. 94B05, 05B35PII. S0895480199364148 Introduction.A linear matroid M together with a chosen representation over a finite field F q is the same object as a linear code. The most well-known result underlining this connection is Greene's theorem on the relation of the weight polynomial of the code and the Tutte polynomial of the matroid. In this paper we further examine the relation between the polynomial invariants of codes, matroids, and some other combinatorial objects. Our point of view is coding-theoretic. We begin with listing basic definitions for linear codes and some very simple linear-algebraic properties of subcodes. These properties lead almost immediately to a relation between the weight polynomial of a linear code and the rank polynomial of the corresponding matroid. This relation is equivalent to Greene's theorem which is shown to be a purely linear-algebraic fact. An advantage of the coding-theoretic point of view is determined by the fact that the weight polynomial enjoys more structural properties than more general matroid invariants; when this structure translates to other problems, it sometimes produces interesting insights.As an example, we relate the reliability polynomial of a linear matroid to an evaluation of the weight polynomial of the code. The corresponding functional on linear codes turns out to be well studied under the name of the probability of undetected error of the code. Together with some related ideas this enables us to derive upper and lower bounds on the matroid reliability. As another application of the weightrank connection, we give a direct proof of the link between the partition function of the Potts model and the weight polynomial of the cocycle code of the graph.General sources for coding theory are the books [17], [19]. Relevant applications of the Tutte polynomial are covered in [6], [24]. All the necessary information on interaction models is contained in [24].

A generalization of weight polynomials to matroids

“…From [9, Example 3] then, we see that the only non-zero Betti numbers of I M(H) (1) are β 0,6 (I M(H) (1) ) = 7 and β 1,7 (I M(H) (1) ) = 6. As always, the (n− r(M (H))− 1) th elongation M (H) (2) has {1, . .…”

“…For example, in addition to the already mentioned equivalence between the Tutte polynomial and the extended weight enumerator of a linear code, it was demonstrated in [3, Theorems 4 and 5] and (independently) in [11,Theorem 3.3.5] that the Tutte polynomial and the set of so-called higher weight enumerators of a linear code determine each other as well. Related results and methods can also be found in [2], where the connection between the weight enumerator and the Tutte polynomial is used to establish bounds on all-terminal reliability of vectorial matroids. In addition, [2] provides new proofs of Greene's theorem itself, and shows how the weight polynomial and the partition polynomial of the Potts model are related.…”

On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

“…The columns of define a -representable matroid on the set of points . For example, we described before the matroid defined by the matrix in (1). This matroid depends only on the code , that is, it does not depend on the choice of the generator matrix .…”

Codes as Fractals and Noncommutative Spaces

“…For a linear code C, one can consider the associated weight polynomial of the code C. We recall here briefly the definition and properties, see [2]. The basic observation is that, for a linear code, The weight polynomial is given by (3.4) A(x, y) = n i=1…”