On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes

@article{Rosnes2012OnTM,
  title={On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes},
  author={Eirik Rosnes and Marcel Ambroze and Martin Tomlinson},
  journal={IEEE Transactions on Information Theory},
  year={2012},
  volume={60},
  pages={5204-5214}
}
In this paper, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m q. In the literature, the minimum/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) has been thoroughly studied for m 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m = 6, the… 

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References

SHOWING 1-10 OF 19 REFERENCES

On the minimum distance of array codes as LDPC codes

This correspondence investigates the minimum distance d(q,j) of the code in an algebraic way and proves that the code is invariant under a doubly transitive group of "affine" permutations.

On the Number of Minimum Stopping Sets and Minimum Codewords of Array LDPC Codes

In this letter, the closed-form formula for the numbers of minimum stopping sets and minimum codewords of C(m,q) are given for 2 ≤ m ≤ 3 and for ( m,q)=(4,5) and (4,7).

On the Stopping Distance of Array Code Parity-Check Matrices

A lower bound on the stopping distances for m > 3 and q > 3 is given and the related minimum distance are determined for one of the given parity check matrices.

More on the Stopping and Minimum Distances of Array Codes

It is shown that in many cases the improper array codes would perform better than the proper array codes over the AWGN and binary erasure channels.

On the minimum weight of simple full-length array LDPC codes

  • K. SugiyamaY. Kaji
  • Computer Science
    2007 IEEE International Symposium on Information Theory
  • 2007
It is shown that the class of SFA-LDPC codes which are denoted by CA (p, 4) contains a codeword whose minimum weight is 10 or less, if p is a prime number greater than 7 and the Yang's lower bound on the minimum weight of CA ( p,4) is exactly 10.

An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices

An efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix is introduced and simulation results of iterative decoding on the binary erasure channel show performance improvements for low-to-medium erasure probabilities when this redundant parity- check matrix is used for decoding.

Addendum to “An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices”

The algorithm for determining the initial part of the stopping set weight spectrum is reviewed, which includes the codeword weight spectrum, and some improvements to the algorithm are provided to provide some improvements.

Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes

Detailed theoretical analysis of fully absorbing sets for the class of Cp, ¿ array-based LDPC codes is provided, including the characterization of all minimal (fully) absorbing sets, and the development of techniques to enumerate them exactly is provided.

Low-density parity-check codes

A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described and the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length.

Analysis of the distribution of the number of erasures correctable by a binary linear code and the link to low-weight codewords

A probabilistic method, which has considerably smaller search space than that of the generator matrix-based methods, is presented to determine the d min of a linear code using random erasure patterns, which shows that the turbo codes that have optimised d min have significantly better performance than LDPC codes.