On the Dimensions of Certain LDPC Codes Based on$q$-Regular Bipartite Graphs

Abstract

Abstract-An explicit construction of a family of binary LDPC codes called LU(3, q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.

Eigenvalues and expansion of bipartite graphs

“…For example, recently, using D(m, q) as the Tanner graph, Kim et al [19] constructed and studied the associated LDPC code LU (m, q). Sin and Xiang [29] determined the dimension of the code LU (3, q) for odd q, and Anslan [2] very recently settled the case of even q.…”

Some low-density parity-check codes derived from finite geometries

“…Over the field G F (2) this was proven in [24]. This result will be extended to all fields with char K = p and for all q, for both LU (3, q) and LU (3, q) D .…”

“…Since LU (3, q) has a square incidence matrix, LU (3, q) and LU (3, q) D have the same dimension. Hence dim C = q 3 − 2q 2 + 3q − 2 2 from [24], since the binary dimension is the same as over any other finite field of characteristic 2. From Theorem 3.3 it follows that this is also equal to dim C , hence dim C = dim C .…”

“…Two structures that have received a lot of attention lately are LU (3, q) and its dual LU (3, q) D [12,13,24]. In [13] the authors conjecture the dimension of the associated code to be…”

Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

“…Recently, q-regular bipartite graphs D(m, q) [12] were used to define the code LU(m, q) [10,20] for the study of LDPC codes. We recall that LU(3, q) has parity check matrix either H (3, q) or H(3, q) T , where H (3, q) is the incidence matrix with rows indexed by lines [x, y, z] ∈ GF(q) 3 and columns indexed by points (a, b, c) ∈ GF(q) 3 .…”