Second-Order Weight Distributions

TL;DR

This paper introduces the second-order weight distribution as a new fundamental property of codes, establishing theoretical results, computing distributions for specific code ensembles, and exploring applications in coding theory and combinatorics.

Contribution

It develops the theory of second-order weight distributions, including MacWilliams identities, and applies it to analyze regular LDPC codes and 2-good random matrices.

Findings

Derived second-order weight distributions for regular LDPC codes

Established MacWilliams identities for second-order weight distributions

Linked second-order weight distributions to properties of 2-good random matrices

Abstract

A fundamental property of codes, the second-order weight distribution, is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles. A series of results, parallel to those for weight distributions, is established for second-order weight distributions. In particular, an analogue of MacWilliams identities is proved. The second-order weight distributions of regular LDPC code ensembles are then computed. As easy consequences, the second moments of weight distributions of regular LDPC code ensembles are obtained. Furthermore, the application of second-order weight distributions in random coding approach is discussed. The second-order weight distributions of the ensembles generated by a so-called 2-good random generator or parity-check matrix are computed, where a 2-good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple-wise properties of sequences. It is shown that the 2-good property is reflected in the second-order weight distribution, which thus plays a fundamental role in some well-known problems in coding theory and combinatorics. An example of linear intersecting codes is finally provided to illustrate this fact.