In-depth introduction to coding theory from both an engineering and mathematical viewpoint.

A complete introduction to the many mathematical tools used to solve practical problems in coding. Mathematicians have been fascinated with the theory of error-correcting codes since the publication of Shannon's classic papers fifty years ago. With the proliferation of communications systems, computers, and digital audio devices that employ error-correcting codes, the theory has taken on practical importance in the solution of coding problems. This solution process requires the use of a wide variety of mathematical tools and an understanding of how to find mathematical techniques to solve applied problems. Introduction to the Theory of Error-Correcting Codes, Third Edition demonstrates this process and prepares students to cope with coding problems. Like its predecessor, which was awarded a three-star rating by the Mathematical Association of America, this updated and expanded edition gives readers a firm grasp of the timeless fundamentals of coding as well as the latest theoretical advances. This new edition features: * A greater emphasis on nonlinear binary codes * An exciting new discussion on the relationship between codes and combinatorial games * Updated and expanded sections on the Vashamov-Gilbert bound, van Lint-Wilson bound, BCH codes, and Reed-Muller codes * Expanded and updated problem sets. Introduction to the Theory of Error-Correcting Codes, Third Edition is the ideal textbook for senior-undergraduate and first-year graduate courses on error-correcting codes in mathematics, computer science, and electrical engineering.

For the practical use of error correcting codes on communications channels, very good, long constructive codes are necessary. The (60, 30) symmetry code over the field of three elements is a long code that has been determined to have the greatest error-correcting properties for a code of length 60 with 30 information symbols over the field of three elements. The (60, 30) symmetry code has more than 10 to the 14th power messages so that determining its weight distribution is a formidable task. In actual use of the code knowledge of the weight distribution is valuable because the weight distribution is required for calculation of the probability of decoding error and failure. In this paper the weight distribution of the (60, 30) symmetry code is given. In connection with this weight distribution, the parameters of some new 5-designs are determined. A conjecture about the entire family of symmetry codes is investigated. (Author).

Join the Cryptokids as they apply basic mathematics to make and break secret codes. This book has many hands-on activities that have been tested in both classrooms and informal settings. Classic coding methods are discussed, such as Caesar, substitution, Vigenère, and multiplicative ciphers as well as the modern RSA. Math topics covered include: - Addition and Subtraction with, negative numbers, decimals, and percentages - Factorization - Modular Arithmetic - Exponentiation - Prime Numbers - Frequency Analysis. The accompanying workbook, The Cryptoclub Workbook: Using Mathematics to Make and Break Secret Codes provides students with problems related to each section to help them master the concepts introduced throughout the book. A PDF version of the workbook is available at no charge on the download tab, a printed workbook is available for $19.95 (K00701). The teacher manual can be requested from the publisher by contacting the Academic Sales Manager, Susie Carlisle

This workbook, which accompanies The Cryptoclub, provides students with problems related to each section to help them master the concepts introduced throughout the book. A PDF version is available at no charge. This file can be found under our Downloads and Updates tab. The teacher manual can be requested from the publisher by contacting the Academ

Self-orthogonal codes have been used on many occasions due to their added ease of decoding. In particular, moderate length self-orthogonal codes are often needed. This paper gives a complete listing of all self-orthogonal codes of rate one-half whose length is between 2 and 18, and also lists all their properties. Someone who wants to use such a code can see what is available from the list and then select one based on the characteristics listed. (Author).

The purpose of this note Is to show that an analogue to Witt's theorem holds for a non-degenerate, non-alternating, symmetric bilinear form f over a field K of characteristic 2 where f(x, x) takes its values in a subfield K* such that K contains the square root of any element in K*. As is known, [2, p. 171] Witt's theorem does not hold in general for a field of characteristic 2. However, the following shows that an isometry of subspace can be extended if it leaves a certain unique vector invariant. The invariants of a subspace of V with respect to the orthogonal group are determined.

The hand calculation of the weight distribution of the (71,35) quadratic residue code is briefly described. This calculation falls into two parts. The first part consists in evaluating the 10x10 determinant given by the powermoment identities, using properties of Van der Monde determinants. The second part of the calculation consists in reducing the six solutions obtained in the first part to one solution. This reduction is accomplished by group theory analysis. (Author).

Witt's theorem is concerned with the extension of an isometry between subspaces to an isometry on the whole space. The most general form of Witt's theorem is theorem 1.2.1 in Wall (3). Theorem 1 of this paper extends theorem 1.2.1 and is identical to it in case the characteristic of the division ring is not 2. Theorem 2 is a variant of theorem 1. Theorems 1 and 2 are concerned with sesquilinear forms. Theorems 3 and 4 are concerned with bilinear forms on a finite dimensional vector space over a field of characteristic 2. Theorem 3 gives necessary and sufficient conditions for two (possibly degenerate) forms to be equivalent. Theorem 4 gives necessary and sufficient conditions for two subspaces to be equivalent.