Box codes and convolutional coding of block codes

Author: Jin, Yonggang

Year: 1995

Degree: Dissertation (Ph.D.)

Advisors: Wilson, Richard M.; Solomon, Gustave

Committee Members: Wilson, Richard M.; Solomon, Gustave; McEliece, Robert J.

Option: Mathematics

DOI: 10.7907/qpt3-fy04

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

Part I.

A self-dual code of length 48, dimension 24, with Hamming distance essentially equal to 12 is constructed. There are only six codewords of weight 8. All the other codewords have weights that are multiples of 4 and have minimum weight equal to 12.

A (72, 36; 15) box code was constructed from a (63, 35; 8) cyclic code. The theoretical justification is presented herein.

A second (72, 36; 15) code is constructed from an inner (63, 27; 16) Bose-Chaudhuri-Hocquenghem (BCH) code and expanded to length 72 using the box code algorithm for extension. This code was simulated and verified to have a minimum distance of 15 with even weight words congruent to 0 modulo 4. The decoding for hard and soft decision is still more complex than the first code constructed above.

Finally, an (8, 4; 5) Reed-Solomon code over GF(512) in the binary representation of the (72, 36; 15) box code gives rise to a (72, 36; 16*) code, where the "16*" means that there are nine codewords of weight 8 and all the rest have weights [...] 16.

Part II.

In order to get self-dual block codes by the convolutional encoding technique developed in [18], Solomon [12] gave sufficient conditions for code length 2n + 2 and tap polynomials [...] and [...]. We present necessary and sufficient conditions for convolutional encoding of self-dual block codes of rate 1/2 with weights [...], [...] 0 (mod 4). In addition [15], we searched for the smallest possible convolutional encoding constraint lengths K for (80, 40; 16) self-dual codes (quadratic residue and non-quadratic residue) and even for (104, 52; 20) quadratic residue code.

Files