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Chinese Science Bulletin

Tight lower bound of consecutive lengths for QC-LDPC codes with girth twelve

Zhang Guo

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in the proofs of the existence of large-girth QC-LDPC codes, the construction of large-girth QC-LDPC codes based on the Chinese remainder theorem, and the construction of LDPC codes with the guaranteed error correction capability.

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IEEE Communications Letters•May 1, 2009•31 citations
For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in the proofs of the existence of large-girth QC-LDPC codes, the construction of large-girth QC-LDPC codes based on the Chinese remainder theorem, and the construction of LDPC codes with the guaranteed error correction capability.
For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in the proofs of the existence of large-girth QC-LDPC codes, the construction of large-girth QC-LDPC codes based on the Chinese remainder theorem, and the construction of LDPC codes with the guaranteed error correction capability.
For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in the proofs of the existence of large-girth QC-LDPC codes, the construction of large-girth QC-LDPC codes based on the Chinese remainder theorem, and the construction of LDPC codes with the guaranteed error correction capability.
For an arbitrary (3, L ) quasi-cyclic(QC) low-density parity-check (LDPC) code with girth at least ten, a tight lower bound of the consecutive lengths is presented. For an arbitrary length above the bound the corresponding LDPC code necessarily has a girth at least ten, and for the length equal to the bound, the resultant code inevitably has a girth smaller than ten. This new conclusion can be well applied to some important issues, such as the proofs of the existence of large girth QC-LDPC codes, the construction of large girth QC-LDPC codes based on the Chinese remainder theorem, as well as the construction of LDPC codes with the guaranteed error correction capability.
A method to construct girth-12 (3,L) quasi-cyclic low-density parity-check (QC-LDPC) codes with all lengths larger than a certain given number is proposed, via a given girth-12 code subjected to some constraints. The lengths of these codes can be arbitrary integers of the form LP, provided that P is larger than a tight lower bound determined by the maximal element within the exponent matrix of the given girth-12 code. By applying the method to the case of row-weight six, we obtained a family of girth-12 (3,6) QC-LDPC codes for arbitrary lengths above 2688, which includes 30 member codes with shorter code lengths compared with the shortest girth-12 (3,6) QC-LDPC codes reported by O'Sullivan.
Enabling the parity-check matrices to have a large girth is an efficient method to improve the decoding performance for many binary and non-binary QC-LDPC codes.A new explicit framework to construct QC-LDPC codes with girth at least eight is proposed,without any computer search procedures.The QC-LDPC codes constructed by the framework not only have a girth at least eight,but also possess an advantage of consecutive cyclic permutation matrix(CPM) sizes.The framework includes two steps:first,a parity-check matrix is explicitly constructed with a girth at least eight under an infinite CPM size;second,the tight lower bound of consecutive CPM sizes is precisely calculated directly from the corresponding shift matrix of the parity-check matrix,by employing a new girth property discovered by this paper.
In this letter, we consider two problems associated with quasi-cyclic low-density parity-check (QC-LDPC) codes. The first is how to extend the code length of a QC-LDPC code without reducing the girth. The second is how to design a QCLDPC code with a prescribed girth easily. We deal with these two problems by using a combining method of QC-LDPC codes via the Chinese Remainder Theorem (CRT). Codes constructed with our proposed method have flexible code lengths, flexible code rates and large girth. Simulation results show that they perform very well with the iterative decoding.