On tensor products of CSS codes

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 $\mathcal{D}$. From this criterion arises a generic bound for the minimum distance which is twice larger than the single bound previously known in the literature. We apply these results 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. More precisely, their minimum distance increases as $O(n^{\alpha })$ for some $\alpha >0$, where $n$ is the code length, while the row weight of their parity–check matrices grows as $O(\log (n))$. This entails a rather surprizing fact: even if a CSS code does not have quantum degeneracy, for a large enough $\ell$, its $\ell$-th iterated tensor power does. Different known results are also reinterpretated in terms of tensor products and three new families of LDPC CSS codes are studied.