CONSTRUCTION OF QUASI-CYCLIC CODES (original) (raw)
The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p Quasi-Cyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some s-th Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting set of circulants for other techniques. Nonbinary Power Residue codes and related QC codes are constructed over GF(3), GF(4), GF(5), GF(7) and GF(8). Their subcodes are also used to find good nonbinary QC codes. A simple and efficient algorithm for constructing primitive polynomials with linearly independent roots over the Galois Field of q elements, GF(q), is developed. Tables of these polynomials are presented. These Tables are unknown for polynomials with nonbinary coefficients, and the known binary Tables are incomplete. The polynomials are employed in such diverse areas as construction of error correcting codes, efficient VLSI implementation of multiplication and inverse operations over Galois Fields, and digital testing of integrated circuits. Using the link established between generalized tail biting convolutional codes and binary QC codes, good QC codes are constructed based on iii Optimum Distance Profile (ODP) convolutional codes. Several best rate 2/3 systematic codes up to circulant size 20 are constructed in this manner.
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