Matrices and Hamming Codes

Abstract

This chapter introduces two Hamming codes, the first modern examples of error correcting codes. A Hamming code provides a way of transforming pieces (words) of a message so that at a later point, a reader of the message will be able to not just detect an error in a word, but correct the error. In order to understand Hamming codes, the chapter begins by introducing some elementary ideas of matrices and linear algebra: row vectors, column vectors and matrices, operations of addition and scalar multiplication, and matrix multiplication. Chapter 3 introduced vectors in the Extended Euclidean Algorithm to find the coefficients in Bezout’s identity and to find integer solutions of integer linear equations in two variables. The idea there was to work with vectors of coefficients of equations that describe successive remainders in Euclid’s Algorithm for two given numbers a and b as integer linear combinations of the two numbers. Matrices play a similar role in isolating and working efficiently with the coefficients of a system of linear equations in order to find solutions of the system. So they will show up again in Chaps. 15, 17 and 19. The chapter ends with a brief description of Hill cryptography, an historically significant generalization to matrices of the multiplicative Caesar cipher introduced in Chap. 2.