C.C. Li 2007/05/24
What I have done this week
1. Survey paper ：
[1] Coset-Based Quasi-Cyclic LDPC Codes for Optimal Overlapped Message
Passing Decoding
In order to achieve the maximum possible throughput gain of the OMP decoding, we propose a coset-based method to construct QC LDPC codes, which can potentially achieve higher throughput gain while maintaining the same theoretical performances as the SFT codes.
Coset-Based Construction
Assume m is prime, then Zm = {0, 1, ・ ・ ・ ,m − 1} is a field under addition and multiplication modulo m, and = {1, 2, ・ ・ ・ ,m − 1} is a multiplicative group under multiplication modulo m. Let a ∈ be an element with multiplicative order k, where k|(m − 1), then <a> = {1, a, a2, ・ ・ ・ , ak − 1} forms a

multiplicative subgroup of , and can be partitioned, with respect to <a>, into disjoint cosets: = , where
Since each coset has k distinct elements, = (m
−
1)/k. By convention, ui is usually called the coset leader of Cui . Any element in Cui can be the leader of the coset, and switching to a different coset leader amounts to cyclically shifting the elements in the coset. Suppose we use as the coset leader of Cui , where 0
≤
k1
≤
k − 1, then we have , . can be obtained by cyclically shifting the elements in Cui to the left by k1 positions.
Any j distinct coset leaders can be used to construct a QC LDPC code: first choose j distinct coset leaders {ui0, ui1 , ・ ・ ・ , uij − 1}, then the corresponding j chosen cosets can be used to define a (j, k) regular QC LDPC code by setting
(mod m) . The resultant parity check matrix is given by
The code constructed based on cosets has length N = km and code rate R
≥
1 − j/k.
BER performance comparisons. For the SFT codes, (a, b) = (16, 33), (39, 19), and
(14, 63) for m = 109, 181, and 397 respectively. For the codes based on cyclotomic

cosets, a = 16, 43, and 79 for m = 109, 181, and 397 respectively, and the coset leaders are {1, 6, 8, 11}, {1, 90, 144, 166} and {1, 20, 133, 306} for m = 109, 181, and 397 respectively. For our cosetbased codes, a = 38, 39, and 14 for m = 109, 181, and 397 respectively, and the coset leaders are {1, 2, 3, 4}, {1, 58, 97, 153} and {1,
59, 111, 355} for m = 109, 181, and 397 respectively.
5/24~5/30

Survey paper about H-QC & QC.