Finite-state codes

A class of codes called finite-state (FS) codes is defined and investigated. The codes, which generalize both block and convolutional codes, are defined by their encoders, which are finite-state machines with parallel inputs and outputs. A family of upper bounds on the free distance of a given FS code is derived. A general construction for FS codes is given, and it is shown that in many cases the FS codes constructed in this way have a free distance that is the largest possible. Catastrophic error propagation (CEP) for FS codes is also discussed. It is found that to avoid CEP one must solve the graph-theoretic problem of finding a uniquely decodable edge labeling of the state diagram

Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes

This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields. Experimental results show that the codes constructed perform well over the binary-input AWGN channel with iterative decoding using the sum-product algorithm (SPA). The second part analyzes the ranks of the parity-check matrices of codes constructed based on finite fields with characteristic of 2 and gives combinatorial expressions for these ranks. The third part identifies a subclass of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this subclass with the SPA converges very fast.Comment: 26 pages, 6 figures, submitted to IEEE Transaction on Communication

Two-dimensional burst identification codes and their use in burst correction

A new class of codes, called burst identification codes, is defined and studied. These codes can be used to determine the patterns of burst errors. Two-dimensional burst correcting codes can be easily constructed from burst identification codes. The resulting class of codes is simple to implement and has lower redundancy than other comparable codes. The results are pertinent to the study of radiation effects on VLSI RAM chips, which can cause two-dimensional bursts of errors

Some Partial Unit Memory Convolutional Codes

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On the existence of optimum cyclic burst-correcting codes

It is shown that for each integer b >= 1 infinitely many optimum cyclic b-burst-correcting codes exist, i.e., codes whose length n, redundancy r, and burst-correcting capability b, satisfy n = 2^{r-b+1} - 1. Some optimum codes for b = 3, 4, and 5 are also studied in detail

Codes for correcting three or more adjacent deletions or insertions

Codes are presented that can correct the deletion or the insertion of a predetermined number of adjacent bits greater than or equal to three. This extends the constructions of codes beyond those proposed by Levenshtein fifty years ago to correct one or two adjacent deletions or insertions