A Construction of Quantum LDPC Codes From Cayley Graphs

High-rate storage codes on triangle-free graphs

“…where the rows of B m are indexed by even-weight vectors of length m − 1, its columns by oddweight vectors, and B m has a 1 in the coordinate indexed by row x and column y if and only if the vector x + y ∈ S m . From the result of [12] quoted before the proposition we find that…”

“…From the point of view of storage, we do not have much use for the quantum code's minimum distance, but we are very much interested in its dimension: in particular we will obtain a storage code of rate greater than 1/2 if and only if the associated quantum code has non-zero dimension. Quantum codes arising from Cayley graphs over F r 2 were studied in [12] and we will make use of some of results obtained in that work.…”

“…We now focus on the case of coset graphs of repetition codes of odd length n = r + 1, since they form a natural generalization of the Clebsch graph which is the coset graph of the 1 Notice that we deal with two types of binary codes: the codes in F n 2 and codes in F N 2 , called small codes and big codes in [12]. [5,4] repetition code.…”

“…, e r denote the canonical generators. Now it turns out that in a quantum coding context, coset graphs G m of repetition codes of even length m were studied in [12], because their adjacency matrices A(G m ) are selforthogonal and therefore directly yield quantum codes. These quantum codes were proved in [12] to have parameters , generator set S m = {e 1 , .…”

“…The codes that we consider are constructed on Cayley graphs on F r 2 (coset graphs of binary linear codes). While coset graphs form a classic topic in coding theory, one motivation of this work is drawn from a recent construction of quantum CSS codes obtained from linear spaces defined by adjacency matrices of coset graphs [12]. Our main results are constructions of two infinite families of binary linear storage codes of rate exceeding 1/2, one of which in fact has rate R 2 (G) approaching 3/4 for growing length N , exceeding the rate 5/8 of the unique previously known binary storage code with rate above 1/2 for triangle-free graphs.…”