LDPC codes from voltage graphs

Abstract

Abstract-Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

On codes designed via algebraic lifts of graphs

“…A similar observation for the theta-graph case is shown in [15]. Moreover, in [5] it is also shown that any such "inevitable cycle" in a derived graph specified by an abelian voltage assignment results from a base graph containing one of these two types of subgraphs. This provides a full classification for the cycles that will always occur in the derived graph when abelian voltages are used (regardless of how they are assigned), due to the base graph structure.…”

“…This suggests that noncommuting voltage assignments have a greater potential in yielding derived graphs with large girth. The result on inevitable cycles in [5] provides an alternative short proof that the girth is always at most 12 for codes constructed using commuting permutation matrices in arrays that contain a sub-array of 2 × 3 nonzero permutation matrices. This limitation of girth 12 was originally shown in [7] and later by various groups (see for example, [17]).…”

“…These are shown in Figure 3. In [5], it was shown that if voltages were assigned from an abelian group G to a base graph G, where G contained either a (a, b, c)-theta graph or a (a 1 , a 2 ; b)-dumbbell graph as a subgraph, 2 A walk W α in the derived graph G α projects down to W if the edges in W α are mapped onto the edges of W by the natural projection mapping in the exact order and orientation of W . then a cycle would inevitably result in the derived graph (i.e.…”

“…This concept, originally coined in topological graph theory [4], may be observed in many well-known families of codes whose underlying graph representations may be interpreted as voltage graphs [5]. For example, quasicyclic LDPC codes based on blocks of shifted-identity matrices, array codes, shortened array codes, quasi-cyclic repeat accumulate codes, and others fall into this category [6], [7], [8], [9], [10], [11].…”

Some voltage graph-based LDPC tailbiting codes with large girth

“…The terminology "base graph" originates from graph theory and is used, for example, in [11]. It differs from the terminology used in [6], [9], where protograph or seed graph are used.…”