A construction of quantum LDPC codes from Cayley graphs

Alain Couvreur (1) , Nicolas Delfosse (2) , Gilles Zemor (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 IMB - Institut de Mathématiques de Bordeaux

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 F_2^n 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(dn^2) 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}]].

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Alain Couvreur, Nicolas Delfosse, Gilles Zemor. A construction of quantum LDPC codes from Cayley graphs. IEEE Transactions on Information Theory, 2013, 59 (9), pp.6087-6098. ⟨hal-00632257v3⟩