Enumerators for Protograph-Based Ensembles of LDPC and Generalized LDPC Codes

Bounds on the Minimum Distance of Punctured Quasi-Cyclic LDPC Codes

“…In fact, the switch from a standard accumulation stage to the jagged accumulation stage, which reduces the number of degree-2 variable nodes, allows the AR4JA protographs to meet the criterion of the ensemble of protographs with linearly increasing minimum distance. More specifically, the techniques of Divsalar and other researchers [10]- [12], [16] can be used to calculate the asymptotic ensemble weight enumerators for protograph-based codes, from which the typical relative minimum distance δ min can be found. They prove that the minimum distance d min of most of the codes in the ensemble derived from the protograph exceeds δ min n, where n is the block length of the code.…”

“…The only subtlety arises from the fact that the sets S excluded from (10) by the min * function may not be excluded from (12) by the application of min * , i.e., there might be sets S such that i∈S wt perm H S\i (x) = 0 but i∈S perm A S\i > 0. The resolution of this potential complication can be achieved by reference to Theorem 8 in [9].…”

“…We note that while our codeword searches generally used error impulse pairs, the codeword of weight 52 at the smallest block length was found using impulse triplets. Divsalar et al showed that most codes in the ensemble of certain protograph-based codes, including the AR4JA codes, have minimum distance linearly increasing with block length [10]- [12]. Specifically, by upper bounding the ensemble average weight enumerator, they were able to prove that Pr {d min < δ min n} → 0 exponentially fast as n → ∞, for some constant δ min > 0.…”

“…In Section VI, we apply our methods to calculate upper bounds on the minimum distance of the codes in the CCSDS standard, which are obtained by a two-step QC expansion of the AR4JA protographs. In Section VII, we use computer search to find low-weight codewords for several AR4JA codes, compare them to our length-independent bounds, and reconcile them with the ensemble minimum-distance lower bounds [10]- [12]. We also examine the girth of AR4JA codes.…”

“…Another line of research has shown that most codes in the ensemble of protograph-based codes characterized by a limited number of degree-two variable nodes have minimum distance that increases linearly with block length [10]- [12]. The family of LDPC codes known as AR4JA codes, recommended for deep-space communications by the Consultative Committee for Space Data Systems (CCSDS) [13], are obtained by puncturing QC-LPDC codes designed from protographs in this ensemble.…”