Distance bounds for generalized bicycle codes

The Generalized Bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices [1][2]. An important advantage of GB codes is that they have a naturally overcomplete set of low-weight stabilizer generators which is expected to improve their performance in the presence of syndrome measurement errors.  Without row weight limit, GB codes with finite encoding rates and distance scaling as Ο(n/log n) exist.  For GB codes with a given row weight w, upper distance bounds can be constructed by mapping them to codes local in D=w-2 dimensions.  We show that several large families of such codes have distances d ≥ O(n^1/2). Numerically, we studied distance scaling in one such family encoding a single qubit, where the distance is seen to grow as a square root of n, with the coefficient increasing with w.

[1] A. A. Kovalev and L. P. Pryadko, "Quantum Kronecker sum-product low-density parity-check codes with finite rate," Phys. Rev. A 88, 012311 (2013)

[2] P. Panteleev and G. Kalachev, "Degenerate quantum LDPC codes with good finite length performance", arXiv:1904.02703 (2019)