A construction of quantum LDPC codes from Cayley graphs

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi in the draft [6]. It is based on the Cayley graph of F₂ⁿ together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn²) where d is the minimum distance of the classical code. When the classical code is the [n, 1, n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2^n-1, 2 n/2, 2 n/2 ^-1]].