Probabilistic bounds on the trapping redundancy of linear codes

TL;DR

This paper establishes a rigorous, sharper upper bound on the trapping redundancy of linear codes, improving previous probabilistic bounds and applicable to a wider range of trapping sets, aiding in code design and analysis.

Contribution

It provides a more general, mathematically rigorous upper bound on trapping redundancy, surpassing previous bounds based on the Lovász Local Lemma, and applicable to all relevant trapping sets.

Findings

The new bound is sharper than previous probabilistic bounds.

It can exactly determine trapping redundancy in many cases.

The bound applies to all potentially avoidable trapping sets with small size.

Abstract

The trapping redundancy of a linear code is the number of rows of a smallest parity-check matrix such that no submatrix forms an $(a,b)$-trapping set. This concept was first introduced in the context of low-density parity-check (LDPC) codes in an attempt to estimate the number of redundant rows in a parity-check matrix suitable for iterative decoding. Essentially the same concepts appear in other contexts as well such as robust syndrome extraction for quantum error correction. Among the known upper bounds on the trapping redundancy, the strongest one was proposed by employing a powerful tool in probabilistic combinatorics, called the Lov\'{a}sz Local Lemma. Unfortunately, the proposed proof invoked this tool in a situation where an assumption made in the lemma does not necessarily hold. Hence, although we do not doubt that nonetheless the proposed bound actually holds, for it to be a mathematical theorem, a more rigorous proof is desired. Another disadvantage of the proposed bound is that it is only applicable to $(a,b)$-trapping sets with rather small $a$. Here, we give a more general and sharper upper bound on trapping redundancy by making mathematically more rigorous use of probabilistic combinatorics without relying on the lemma. Our bound is applicable to all potentially avoidable $(a,b)$-trapping sets with $a$ smaller than the minimum distance of a given linear code, while being generally much sharper than the bound through the Lov\'{a}sz Local Lemma. In fact, our upper bound is sharp enough to exactly determine the trapping redundancy for many cases, thereby providing precise knowledge in the form of a more general bound with mathematical rigor.