An error-correcting code may be thought of as a subset S_0 of points belonging to a set S in which a metric is defined such that the distance between every pair of distinct points of S_0 is larger than some given number. In Hamming's original formulation, S was taken to be the set of all 2^n n -bit binary numbers and the distance between a pair of binary numbers s and t was taken to be the number of bits of s which do not agree with the corresponding bits of t . In this note we shall take S to be the set of all n -tuples in which each coordinate of an n -tuple can assume one of k integral values: 0, 1, \cdots, k - 1 , with k \geqq 2 . Properties of these nonbinary codes will be discussed.

Some properties of nonbinary error-correcting codes

It is proved that there are no unknown perfect (Hamming-)error-correcting codes over finite fields.

On the Nonexistence of Perfect Codes over Finite Fields

/pdf/mathematical-problems-in-the-shannon-theory-of-optimal-cnf4ox90w8.pdf

Mathematical Problems in the Shannon Theory of Optimal Coding of Information

Vasiliev's method Vasiliev 1963 for the construction of linear and nonlinear, single error-correcting binary perfect codes is generalized to every prime power q.

On linear and nonlinear single-error-correcting q-nary perfect codes

On tactical configurations and error-correcting codes*

Hamming considered the problem of efficient, faultless transmission of binary data over a noisy channel. For a channel which corrupts no more than one binary digit in each sequence of length n, he constructed alphabets, the so-called Hamming codes, which permit error-free signalling. The authors study the analogous problem for channels which can corrupt a greater number of digits. Non-binary channels are also studied, and analogues of the Hamming codes are constructed. It is perhaps of interest that some of the techniques employed derive from algebraic and analytic number theory, mathematical disciplines not generally associated with the type of applied problems considered in this paper.

D. L. Slotnick

Papers

On the mathematical theory of error-correcting codes