On tensor products of CSS Codes

TL;DR

This paper explores tensor products of CSS codes, establishing bounds on minimum distances, analyzing iterated tensor powers, and introducing new code families with favorable properties like logarithmic LDPC structure and high minimum distances.

Contribution

It provides a criterion for minimum distance bounds under tensor products, studies iterated tensor powers, and introduces three new CSS code families with improved parameters.

Findings

Minimum distances of tensor powers tend to infinity.

New CSS code families with logarithmic row weight and large minimum distances.

Reinterpretation of known results using tensor product framework.

Abstract

CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product $\otimes$ which induces a similar operation on the former. We investigate this operation, and in particular its behavior with regard to minimum distances. Given a CSS code $\mathcal{C}$, we give a criterion which provides a lower bound on the minimum distance of $\mathcal{C} \otimes \mathcal{D}$ for every CSS code $\mathcal D$. We apply this result to study the behaviour of iterated tensor powers of codes. Such sequences of codes are logarithmically LDPC and we prove in particular that their minimum distances tend generically to infinity. Different known results are reinterpretated in terms of tensor products. Three new families of CSS codes are defined, and their iterated tensor powers produce LDPC sequences of codes with length $n$, row weight in $O(\log n)$ and minimum distances larger than $n^{\frac{\alpha}{2}}$ for any $\alpha<1$. One family produces sequences with dimensions larger than $n^\beta$ for any $\beta<1$.