More Sparse-Graph Codes for Quantum Error-Correction

We use Cayley graphs to construct several dual-containing codes, all of which have sparse graphs. These codes’ properties are promising compared to other quantum error-correcting codes. This paper builds on the ideas of the earlier paper Sparse-Graph Codes for Quantum ErrorCorrection (quant-ph/0304161), which the reader is encouraged to refer to. To recap: Our aim is to make classical error-correcting codes with practical potential for quantum error-correction. The rules of the game for classical binary codes are: (1) each code must be defined by a sparse graph (so that the circuit for computing the syndrome is simple, and so that we have the chance of getting good decoding with a low-complexity decoder); (2) the classical code must contain its own dual; equivalently, every row of the parity check matrix must be a codeword; equivalently, any two rows of the parity check matrix must have even overlap. (3) the code must have rate greater than 1/2. While struggling to make codes satisfying these constraints, we formulated two mutuallyexclusive conjectures 1 Conjecture G: Any dual-containing code defined by an M×N parity check matrix H with M < N/2, all of whose rows have weight ≤ k, has codewords of weight ≤ k that are not in the dual. Conjecture D: There exist dual-containing codes with sparse parity-check matrix and good distance. To be precise, such codes would have a parity-check matrix with maximum row weight k, and for increasing blocklength N the minimum distance d of codewords not in the dual would satisfy d ∝ N . In this paper we present some more algebraic constructions of binary codes that contain their duals. These codes have the nice property that their sparse parity check matrices have many redundant rows. (These redundant rows allow enhanced decoding performance.) We have found codes that are counterexamples to Conjecture G. 1 Cayley-graph construction of a parity-check matrix Given a set of k generators of a group of size N , we can make a bipartite graph with N vertices on each side and degree k by putting an edge from each group element (vertex on the left) to each group element (vertex on the right) that can be reached by applying one of the generators. If the inverses of the generators are in the set of generators, if the group is abelian, and if k is even, then the graph will be symmetric and will define a square paritycheck matrix with the property that the overlap between any two rows is even. This will thus define a code that is dual-containing. An example. Throughout this paper, N will be a power of 2 (N = 2n), and the group elements are the set {0, 1}n. Our method is easiest to describe if we number our vertices from 0 to N − 1. A vertex (described by an integer in (0,N − 1)) should be thought of as defining a binary string of length n. Input: A set of k integers g_1 ... g_k (the generators) 1. Create an N x N matrix consisting of zeros. 2. for n = 0 ... N-1 do for g in {g_1 .. g_k} do set entry [n, n^g] to 1 (where ^ denotes exclusive or)