Coding theory MOC

Code

A -ary code of length is a inhabited subset , where is a set (called an alphabet) containing letters.¹ code

• An element is thence called a codeword.
• The Hamming distance between codewords is the number of positions in which they differ, and makes a metric space.
• The weight of a code is the distance from the zero-codeword , where consists of some distinguished letter .

Following van Lint, a code if length with codewords and ^minimumDistance is called an -code. An important special case is a linear code, where we take , the Galois field of order , and require to be a vector subspace.

Further notions

• The minimum distance of a non-unary code is

• The minimum weight of a non-unary code is
• The information rate of a -ary code of length is
• The covering radius of a a code is the minimum radius required for Hamming balls around codewords to cover the whole space, i.e.
• Equivalence of codes

Special kinds of code

• Linear code

• Perfect code

• Systematic code

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Footnotes

1. 1999. Introduction to coding theory, §3.1, pp. 33–34 ↩