A Construction of Quantum LDPC Codes from Cayley Graphs

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We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn 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 [[2ⁿ, 2ⁿ⁺¹/², 2ⁿ⁻¹/²]].