We look at low density parity check codes over a finite field K associated with finite geometries T ∗ 2 (K), where K is any subset of PG(2, q), with q = ph, p 6= charK. This includes the geometry LU(3, q)D, the generalized quadrangle T ∗ 2 (K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minim...

Abstract— The problem of designing codes with close-tocapacity performance on the multiple-input multiple-output (MIMO) quasi-static fading channel (QSFC) is addressed. We consider three different coding schemes – namely, direct transmission of random-like codes, concatenation with linear processing orthogonal space-time block codes (o-STBC) and concatenation with space-time trellis codes (STTC...

We show how expander based arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages. The implication of this result is that when the block length is sufficiently large, once a message passing algorithm has corrected a sufficiently large fraction of the errors, it will eventually correct all errors. This result is then combined with known r...

We present simple constructions of optimal erasure-correcting LRC codes by exhibiting their parity-check matrices. When the number of local parities in a parity group plus the number of global parities is smaller than the size of the parity group, the constructed codes are optimal with a field of size at least the length of the code. We can reduce the size of the field to at least the size of t...

Low-density parity-check (LDPC) codes are very powerful error-correction codes with capabilities approaching the Shannon’s limits. In evaluating the error performance of a LDPC code, the computer simulation time taken becomes a primary concern when tens of millions of noise-corrupted codewords are to be decoded, particularly for codes with very long lengths. In this paper, we propose modeling t...

Using group theory, we analyze cycle GF(2) codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of ...

42 JANUARY 2004 1053-5888/04/$20.00©2004IEEE e consider the problem of designing unoriented bipartite graphs with large girth. These graphs are the Tanner graphs associated with the parity-check matrix H of low density parity-check (LDPC) codes or Gallager codes. Larger girth improves the computational and bit error rate (BER) performance of these codes. The article overviews several existing m...

Using group theory, we analyze cycle GF(2) codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of ...