Unit Memory Convolutional Codes Research Articles

Improved Block Oriented Unit Memory Convolutional Codes

This paper is concerned with a special class of unit memory convolutional codes (UMCCs), called block oriented UMCCs (BOUMCCs). Distinguished from conventional UMCCs, which usually have small constraint lengths, the BOUMCCs have relatively large constraint lengths. We conduct the performance analysis by assuming a first-order Markov model, which indicates that the performance of the BOUMCCs depends critically on both the error propagation and the sub-frame error rate of the first layer. The error propagation can be alleviated by the use of partial superposition, which is specified by a superposition matrix with a fraction of columns being nulled. Given a superposition fraction, we propose a tree growing and pruning algorithm (TGPA) with a tunable sliding window, which provides a convenient way to trade off the decoding delay and the performance. We also present a structured construction and show by simulation that there is no performance degradation compared with random construction. Numerical results also show that, by taking the TBCCs as basic codes, the performance of BOUMCCs with TGPA is comparable to that of other short codes but with a more flexible construction or a lower complexity.

A class of new group character codes and convolutional codes

In this paper, we construct a class of new group character codes with parameters \([2^n,2^n-2S_n(r-1),2^r]\) based on Ding et al. (IEEE Trans Inf Theory 46(1):280–284, 2000). Moreover, we use the new group character codes to construct a family of unit memory convolutional codes with parameters \([2^n,2^n-S_n(r-1),S_n(r-1);1,2^r]\), which the free distances of these convolutional codes are exact values.

Multiscale unit‐memory convolutional codes

In this study, authors propose structure to unit memory (UM) convolutional codes that greatly simplify their decoding when using sequential or suboptimal L-decoders. The multiscale structure proposed in this study preserves the word-oriented nature of unit memory codes (UMCs), but allows for processing the code in blocks of sizes less than the memory of the code with non-Viterbi-type decoders. At one end of the multiscale structure are generic UMCs. At the other end are structured systematic UMCs that can be decoded with the same ease as conventional multimemory convolutional codes with sequential or L-decoders. As a special case, the authors show quick-look parity check systematic codes of arbitrary rates that are better than previously known optimum minimum distance systematic convolutional codes. These codes are derived from block codes and the convolutional code can be decoded using the syndrome decoder of the underlying block code. The authors present simulation results and analysis to support the proposed multiscale UM convolutional codes.

A repeat request strategy based on sliding window decoding of unit-memory convolutional codes

In this correspondence, we investigate a decision feedback strategy for convolutional codes which is based on a sliding window decoding procedure and a threshold test as decision rule. For this purpose, we introduce the burst distance spectrum of a convolutional code and derive asymptotic bounds for the ensemble of periodically time-varying convolutional codes. These results are helpful for the asymptotic analysis of the decision feedback scheme. We show that unit memory codes are particularly suited for such a transmission scheme. For these codes, the decoding procedure is reduced to the decoding of block codes with lengths in the order of the overall constraint length of the convolutional code. This leads to a significantly smaller decoding complexity compared with other known decoding and decision rules. Whereas the achievable asymptotic performance is close to the best known bounds. For low rates, our results even improve these bounds.

Active distances for convolutional codes

A family of active distance measures for general convolutional codes is defined. These distances are generalizations of the extended distances introduced by Thommesen and Justesen (1983) for unit memory convolutional codes. It is shown that the error correcting capability of a convolutional code is determined by the active distances. The ensemble of periodically time-varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble. The active distances are very useful in the analysis of concatenated convolutional encoders.

A link between quasi-cyclic codes and convolutional codes

Given a quasi-cyclic code with minimum Hamming distance d, a set of convolutional codes is derived with free distance equal to d. It is shown that an increase in the rate of these codes results in a decrease in the memory length. The connection between these codes is illustrated with several well-known quasi-cyclic codes. The free distance of some partial unit memory convolutional codes can be determined using the results in this correspondence.

Two-step trellis decoding of partial unit memory convolutional codes

We present a new soft-decision decoding method for high-rate convolutional codes. The decoding method is especially well-suited for PUM convolutional codes. The method exploits the linearity of the parallel transitions in the trellis associated with PUM codes. We provide bounds on the number of operations per decoded bit, and show that this number is dependent on the weight hierarchy of the linear block code associated with the parallel transitions. The complexity of the new decoding method for PUM codes is compared to the complexity of Viterbi decoding of comparable punctured convolutional codes. Examples from a special class of PUM codes show that the new decoding method compares favorably to Viterbi decoding of punctured codes.

A directed search approach for unit-memory convolutional codes

A set of heuristic algorithms to numerically search for binary unit-memory convolutional codes (UMC) are presented along with a large number of new codes for 2/spl les/k/spl les/8 and code rate 1/4/spl les/R<1. Combinatorial optimization is used which involves selecting and then pairwise-matching column vectors of the two (n,k) UMC tap weight matrices. The column selection problem is that of finding the best (2n,k) binary, linear block code (BC). In this correspondence, the best BC generator matrix G is found by successively refining G using directed local exhaustive searches. In particular, the set of minimum-weight codewords are used to find a subset of G to exhaustively search. The UMC search strategy (pairwise matching problem) uses a directed local exhaustive search similar to the BC directed search by using the concept of the terminated BC of the UMC. The heuristic algorithms developed in this correspondence are very robust and converge relatively quickly to the optimal or near-optimal UMC. In addition, although it is generally possible to achieve the block code upper bound for free distance, we give a class of UMCs which cannot achieve this bound.

Using combinatorial optimization to design good unit-memory convolutional codes

A method for designing good unit-memory convolutional codes is presented. The method is based on the decomposition of the original problem into two easier subproblems that can be formulated as optimization problems and solved by efficient heuristic search algorithms. The efficacy of this method is demonstrated by a table containing 33 new unit-memory convolutional codes (n,k) with 5<or=k<or=8, rates R=k/n between 1/4 and 7/9, and complete memory (M=k), as well as 12 new linear block codes. Most of the codes found have minimum free distance.<<ETX>>

Quasi-cyclic unit memory convolutional codes

Unit memory convolutional codes with generator matrices, which are composed of circulant submatrices, are introduced. This structure facilitates the analysis of efficient search for good codes. Equivalences among such codes and some of the basic structural properties are discussed. In particular, catastrophic encoders and minimal encoders are characterized and dual codes treated. Further, various distance measures are discussed, and a number of good codes, some of which result from efficient computer search and some of which result from known block codes, are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>