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Advances and Applications in Discrete Mathematics

OPTIMIZED MULTI-RATE QC-LDPC CODE CONSTRUCTION USING FINITE EUCLIDEAN GEOMETRY

Gurinderjit Kaur, Manish Gupta, Rajesh Kumar Narula, Jaskarn Singh Bhullar

Published Jul 30, 2026• 0 citations

Abstract

Finite Euclidean Geometries (EG) are mathematical structures that generalize the familiar concepts of Euclidean geometry to a finite field. In the context of error-correcting codes, these geometries provide a structured way to construct multi-rate quasi-cyclic low-density parity-check codes. Firstly, an algebraic approach is proposed to obtain an exponent matrix, secondly the bottom edge segmentation (BES) algorithm is implemented to get the code of girth 8 and 10. The bit error rate of the proposed code at SNR between 4 dB to 5 dB is $10^{-10}$. In this paper, the code gain has been improved from 0.5 dB to 3.0 dB at $10^{-4}$ BER for 16200 bit code from 0.5 dB to 2.0 dB. For a code length of 32400 bits, BER levels below $10^{-9}$ are achieved with a gap to the Shannon limit ranging from 1.287 dB to 2.603 dB. The sum-product algorithm is employed to reduce the decoding threshold. The proposed multi-rate quasi-cyclic low-density parity-check codes suit 5G wireless communication systems, cloud data storage systems, and satellite communication systems.

DOI: 10.17654/0974165826058

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Finite Euclidean Geometries (EG) are mathematical structures that generalize the familiar concepts of Euclidean geometry to a finite field. In the context of error-correcting codes, these geometries provide a structured way to construct multi-rate quasi-cyclic low-density parity-check codes. Firstly, an algebraic approach is proposed to obtain an exponent matrix, secondly the bottom edge segmentation (BES) algorithm is implemented to get the code of girth 8 and 10. The bit error rate of the proposed code at SNR between 4 dB to 5 dB is

10^{-10}

. In this paper, the code gain has been improved from 0.5 dB to 3.0 dB at

10^{-4}

BER for 16200 bit code from 0.5 dB to 2.0 dB. For a code length of 32400 bits, BER levels below

10^{-9}

are achieved with a gap to the Shannon limit ranging from 1.287 dB to 2.603 dB. The sum-product algorithm is employed to reduce the decoding threshold. The proposed multi-rate quasi-cyclic low-density parity-check codes suit 5G wireless communication systems, cloud data storage systems, and satellite communication systems.

Low density parity check (LDPC) codes with iterative decoding based on belief propagation (IDBP) achieve astonishing error performance close to the Shannon limit. Until now there has been no known method for constructing these Shannon limit approaching codes systematically. Good LDPC codes are largely generated by computer search. As a result, the encoding of long LDPC codes is in general very complex. This paper presents the first algebraic method for constructing LDPC codes systematically based on finite analytic geometries. Four classes of finite geometry LDPC codes with relatively good minimum distances are constructed. These codes are either cyclic or quasi-cyclic and therefore their encoding can be implemented with simple linear feedback shift registers. Long finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.

This paper proposes a new technique for constructing the Quasi-Cyclic low density parity check (QC-LDPC) codes based on row division method. The new codes offer more flexibility in term of girth, code rates and codeword length. In this method of code construction, the rows are used to form as the distance graph. Then they are transformed to a parity-check matrix in order to acquire the desired girth. The new QC-LDPC codes show good bit-error rate performance as compared to the renowned Mackay codes for given values of Eb/Noby 0.12 dB at BER of10-7.

This letter presents an approach to the construction of multiple-rate quasi-cyclic (QC) low-density parity-check (LDPC) codes based on hyperplanes (μ-flats) of two different dimensions in Euclidean geometries. The codes constructed with this method have the same code length, multiple-rate and large stopping sets while maintaining the same basic hardware architecture. The code performance is investigated in terms of the bit error rate (BER) and compared with those of the LDPC codes which are proposed in IEEE 802.16e standard. Simulation results show that our codes perform very well and have low error floors over the AWGN channel.

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This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.