Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes

Abstract

Abstract-In this paper, we study the design of nonbinary low-density parity-check (LDPC) cycle codes over Galois field GF(q). First, we construct a special class of nonbinary LDPC cycle codes with low error floors. Our construction utilizes the cycle elimination algorithm to remove short cycles in the normal graph and to select nonzero elements in the parity-check matrix to reduce the number of low-weight codewords generated by short cycles. Furthermore, we show that simple modifications of such codes are parallel sparse encodable (PSE). The PSE code, consisting of a quasi-cyclic (QC) LDPC cycle code and a simple tree code, has the attractive feature that it is not only linearly encodable, but also allows parallel encoding which can reduce the encoding time significantly. We provide a systematic comparison between nonbinary coded systems and binary coded systems. For the MIMO channel considered, our results show that the proposed nonbinary system employing the PSE code outperforms not only the binary LDPC code specified in the 802.16e standard, but also the optimized binary LDPC code obtained using the EXIT chart methods.

Nonbinary Quasi-Regular QC-LDPC Codes Derived From Cycle Codes

“…(7) and (8)]) and (175,125) codes from (7,5)-cage ([3, eqs. (9) and (10)]). The algorithm indirectly optimizes degree distribution of the code, represented by the optimized average column weight η, which is also shown in Tab.…”

“…Given a binary H as in (1), a nonbinary structured matrix over GF(q) can be obtained by substituting nonzero entries in H with elements from GF(q). If the resultant code is required to be Quasi-Cyclic, a special constraint should be imposed on the selection of the nonzero entries, specifically in every circulant row, the nonzero element should be α λ -multiplied element of the row above it [9]. Let the code graph corresponding to H be G = (V, E), where V is the set of nodes including check nodes and symbol …”

“…Construct code over GF(q) from (V, E) and nonzero elements generated with some known method, e.g. [9], [10], forcing α λ -multiplied circulants [9], if the Quasi-Cyclic feature is required. Simulate performance of the new NB-LDPC code.…”

Computing the Minimum Distance of Nonbinary LDPC Codes

“…• Reference [28] generalized the approach in [27] to a β-multiplied circulant permutation matrix which translates to L|(q − 1) and λ = (q − 1)/L. More generally, here following [29] we select λ once L is given.…”

Design of Binary and Nonbinary Codes from Lifting of Girth-8 Cycle Codes with Minimum Lengths

“…The encoding and decoding operations of cycle codes have lower computation, storage complexity and the block error statistic of these codes is more compatible with outer Reed-Solomon codes, compared to LDPC codes with column-weight at least three [4]. Also, it has been shown [6], [7] that designing cycle codes with large girths, especially in the non-binary setting [8]- [11], is highly beneficial for the error floor performance. Construction of cycle codes with large girth has been investigated by several authors.…”