Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory - developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.

Designs and their codes

This paper presents some basic theorems giving the structure of cyclic codes of lengthn over the ring of integers modulop

 a
 and over thep-adic numbers, wherep is a prime not dividingn. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomialX

 3
 
+λX

 2
 
+(λ−1)X−1, where λ satisfies λ2 - λ + 2 = 0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adic Golay code of length 24 and the 3-adic Golay code of length 12.

/pdf/modular-and-p-adic-cyclic-codes-1wicxp4ats.pdf

Modular and p -adic cyclic codes

https://cacr.uwaterloo.ca/techreports/2006/cacr2006-10.pdf

On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

A Swan-like theorem

This article surveys some selected topics in algebraic coding theory and their links to geometry and cryptography.

Topics in Algebraic Coding Theory

On minimum weight codewords in QR codes

Swan's theorem for binary tetranomials

A binary polynomial p(x) will be said to be of tetranomial type precisely when (x + l)p(x) has just four terms. One motivation for this is the common use of tetranomials of the form (x + l)p(x), where p(x) is primitive, for cyclic redundancy-check codes (CRC’s); in addition, such polynomials share a simple pattern, namely that all the zero coefficients occur consecutively. This paper analyzes the distribution and algebraic properties of irreducibles of tetranomial type, with particular attention to comparisons with irreducible trinomials. A crucial part of the analysis is an extension of Swan’s Trinomial Theorem to binary tetranomials.

Irreducibles of Tetranomial Type

A method of correcting a communication signal with BCH product codes is disclosed. The method comprising the steps of receiving a codeword vector, establishing a generator polynomial, establishing a check polynomial, calculating a binary-matrix, and calculating the binary syndrome S=Hr t . If an all zero vector results, no errors exist in the received vector. If errors exist, a tm x (N−k) binary matrix is created, the power basis expansion of (r(α 1 ), r(α 3 ), r(α 5 ), . . . r(α 2t−1 )) is calculated, and the syndrome is solved. The speed of the method comes from using word-level XOR's to apply the check polynomial to the stream at all offsets. The utility of this approach is due in part to the fact that the conversion matrix involved can be created quickly using only items needed elsewhere in BCH decoders: a field table and the generator polynomial coefficients.

Donald W. Newhart

Papers

On minimum weight codewords in QR codes

Swan's theorem for binary tetranomials

Irreducibles of Tetranomial Type

Method of decoding signals having binary BCH codes

Information sets in quadratic-residue codes