Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields

2011 ◽  
Vol 59 (9) ◽  
pp. 2330-2336 ◽  
Author(s):  
Li Zhang ◽  
Shu Lin ◽  
Khaled Abdel-Ghaffar ◽  
Zhi Ding ◽  
Bo Zhou
Keyword(s):  
Finite Fields ◽  
Ldpc Codes ◽  
Cyclic Subgroups

scholarly journals THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

2020 ◽  
Vol 71 (3) ◽  
pp. 781-822
Author(s):  
Corentin Perret-Gentil
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Divisor Function ◽  
Finite Abelian Groups ◽  
Cyclic Subgroups ◽  
Number Of Divisors ◽  
Curves Over Finite Fields ◽  
Euler Products

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

scholarly journals Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hengzhou Xu ◽  
Huaan Li ◽  
Jixun Gao ◽  
Guixiang Zhang ◽  
Hai Zhu ◽  
...  
Keyword(s):  
Finite Fields ◽  
Gaussian Noise ◽  
Iterative Decoding ◽  
Additive White Gaussian Noise ◽  
Ldpc Codes ◽  
Parity Check ◽  
Undirected Graphs ◽  
Awgn Channel ◽  
Exponent Matrix ◽  
Large Girth

In this paper, we study a class of nonbinary LDPC (NBLDPC) codes whose parity-check matrices have column weight 2, called NBLDPC cycle codes. We propose a design framework of 2 , ρ -regular binary quasi-cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and finite fields by randomly choosing the nonzero field elements in their parity-check matrices. For enlarging the girth values, our approach is twofold. First, we give an exhaustive search of circulants with column/row weight ρ and design a masking matrix with good cycle distribution based on the edge-node relation in undirected graphs. Second, according to the designed masking matrix, we construct the exponent matrix based on finite fields. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are finally provided.

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