Special sequences as subcodes of reed-solomon codes

We consider sequences in which every symbol of an alphabet occurs at most once. We construct families of such sequences as nonlinear subcodes of a q-ary [n, k, n − k + 1] _q Reed-Solomon code of length n ≤ q consisting of words that have no identical symbols. We introduce the notion of a bunch of words of a linear code. For dimensions k ≤ 3 we obtain constructive lower estimates (tight bounds in a number of cases) on the maximum cardinality of a subcode for various n and q, and construct subsets of words meeting these estimates and bounds. We define codes with words that have no identical symbols, observe their relation to permutation codes, and state an optimization problem for them.     

Bounds on the minimum code distance for nonbinary codes based on bipartite graphs

The minimum distance of codes on bipartite graphs (BG codes) over GF(q) is studied. A new upper bound on the minimum distance of BG codes is derived. The bound is shown to lie below the Gilbert-Varshamov bound when q ≤ 32. Since the codes based on bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ≤ 32) BG codes are worse than the best known linear codes. This is the key result of the work. We also obtain a lower bound on the minimum distance of BG codes with a Reed-Solomon constituent code and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed-Solomon constituent code. The bound for LDPC codes is very close to the Gilbert-Varshamov bound and lies above the upper bound for BG codes.     

Generalized error-locating codes with component codes over the same alphabet

We consider generalized error locating (GEL) codes over the same alphabet for both component codes. We propose an algorithm for computing an upper bound on the decoding error probability under known input symbol error rate and code parameters. Is is used to construct an algorithm for selecting code parameters to maximize the code rate for a given construction and given input and output error probabilities. A lower bound on the decoding error probability is given. Examples of plots of decoding error probability versus input symbol error rate are given, and their behavior is explained.     

Low-density parity-check codes based on Galois fields

Methods for constructing a mapping of the elements of a multiplicative group of a Galois field onto a symmetric group of permutation matrices are proposed. A technique minimizing the order of the symmetric group is suggested. The results are used for constructing an ensemble of low-density parity-check codes. The obtained code constructions are tested on an iterative belief propagation (sum-product) decoding algorithm on transmission of a code word through a binary channel with an additive Gaussian white noise.     

On the active distances of partial unit memory codes based on LDPC codes

Partial unit memory (PUM) convolutional codes constructed from block codes make it possible to obtain convolutional codes with high correcting properties. An ensemble of binary PUM convolutional codes based on low-density parity-check block codes is introduced. This ensemble is specified by semi-infinite parity-check matrices with partial unit memory. The asymptotic bounds on a free distance and active row distances are derived for codes of the specified ensemble. It is proved that the ensemble codes have the positive slope of active row distances.     

Woven codes with outer warp: variations, design, and distanceproperties

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties     

Erasure correction by low-density codes

We generalize the method for computing the number of errors correctable by a low-density parity-check (LDPC) code in a binary symmetric channel, which was proposed by V.V. Zyablov and M.S. Pinsker in 1975. This method is for the first time applied for computing the fraction of guaranteed correctable erasures for an LDPC code with a given constituent code used in an erasure channel. Unlike previously known combinatorial methods for computing the fraction of correctable erasures, this method is based on the theory of generating functions, which allows us to obtain more precise results and unify the computation method for various constituent codes of a regular LDPC code. We also show that there exist an LDPC code with a given constituent code which, when decoded with a low-complexity iterative algorithm, is capable of correcting any erasure pattern with a number of erasures that grows linearly with the code length. The number of decoding iterations, required to correct the erasures, is a logarithmic function of the code length. We make comparative analysis of various numerical results obtained by various computation methods for certain parameters of an LDPC code with a constituent single-parity-check or Hamming code.     

Error exponents for product convolutional codes

An upper bound on the error probability (first error event) of product convolutional codes over a memoryless binary symmetric channel, and the resulting error exponent are derived. The error exponent is estimated for two decoding procedures. It is shown that, for both decoding methods, the error probability exponentially decreasing with the constraint length of product convolutional codes can be attained with nonexponentially increasing decoding complexity. Both estimated error exponents are similar to those for woven convolutional codes with outer and inner warp.