A construction of quantum LDPC codes from Cayley graphs

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

1 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 Fn 2 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}]].

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. 2011. ⟨hal-00632257v1⟩