TY - JOUR

T1 - The Leech Lattice and the Golay Code

T2 - Bounded-Distance Decoding and Multilevel Constructions

AU - Amrani, Ofer

AU - Be'ery, Yair

AU - Vardy, Alexander

AU - van Tilborg, Henk C.A.

AU - Sun, Feng Wen

N1 - Funding Information:
Manuscript received January 25, 1993; revised August 25, 1993. The research of A. Vardy was supported in part by the Eshkol Fellowship administered by the Israel Ministry of Science and in part by the Rothschild Fellowship administered by the Rothschild Yad Hanadiv Foundation. This paper is a combination of the results of independent studies by 0. Amrani, Y. Be’ery, and A. Vardy, “Reducedcomplexity bounded-distance decoding of the Leech lattice,” and by F.-W. Sun and H. C. A. van Tilborg, “Multilevel construction of the Golay code and the associated soft-decision decoding algorithm.” This paper was presented in part at the IEEE International Symposium on Information Theory, San Antonio, TX,J anuary 17-21, 1993, and at the Information Theory Workshop (ITW-931, Shizuoka, Japan, June 1993.

PY - 1994/7

Y1 - 1994/7

N2 - Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the projection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructions, the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates. Based upon the multilevel structure of the Golay code and the Leech lattice, efficient bounded-distance decoding algorithms are devised. The bounded-distance decoder for the binary Golay code requires at most 431 operations, as compared to 651 operations for the best known maximum-likelihood decoder. Efficient bounded-distance decoding of the Leech lattice is achieved by means of partitioning it into four cosets of Q24, beyond the conventional partition into two H24cosets. The complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case, as compared to about 3600 operations for the best known maximum-likelihood decoder. It is shown that the proposed algorithms decode correctly at least up to the guaranteed error-correction radius of the maximum-likelihood decoder. Thus, the loss in coding-gain is due primarily to an increase in the effective error-coefficient, which is calculated exactly for both algorithms. Furthermore, the performance of the Leech lattice decoder on the AWGN channel is evaluated experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10-1to 10-7.

AB - Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the projection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructions, the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates. Based upon the multilevel structure of the Golay code and the Leech lattice, efficient bounded-distance decoding algorithms are devised. The bounded-distance decoder for the binary Golay code requires at most 431 operations, as compared to 651 operations for the best known maximum-likelihood decoder. Efficient bounded-distance decoding of the Leech lattice is achieved by means of partitioning it into four cosets of Q24, beyond the conventional partition into two H24cosets. The complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case, as compared to about 3600 operations for the best known maximum-likelihood decoder. It is shown that the proposed algorithms decode correctly at least up to the guaranteed error-correction radius of the maximum-likelihood decoder. Thus, the loss in coding-gain is due primarily to an increase in the effective error-coefficient, which is calculated exactly for both algorithms. Furthermore, the performance of the Leech lattice decoder on the AWGN channel is evaluated experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10-1to 10-7.

KW - Golay code

KW - Leech lattice

KW - bounded-distance decoding

KW - multilevel constructions

UR - http://www.scopus.com/inward/record.url?scp=0028461652&partnerID=8YFLogxK

U2 - 10.1109/18.335970

DO - 10.1109/18.335970

M3 - מאמר

AN - SCOPUS:0028461652

VL - 40

SP - 1030

EP - 1043

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 4

ER -