Coding theory MOC

Linear code

A -ary linear code of length and dimension is a -dimensional vector subspace , code where is the Galois field of order . Thus it is a particular kind of code of length in the alphabet . Following van Lint, a -dimensional linear code of length and ^minimumDistance is called an code, where the is optional.¹ Of particular interest are binary linear codes.

Further notions

• A code is degenerate iff some digit is zero for all codewords.

• is equipped with a natural ^nondegenerate ^symmetric bilinear form

  which is used to define the Orthogonal code.

• A generator matrix has as its row space, and is said to be in standard form iff it is in reduced row echelon form . The first digits are thence information digits and the latter are parity check digits. Every code is equivalent to one generated by such a standard form matrix.

• The generator matrix of the Orthogonal code is called the parity check matrix, since .

• The value of is called the syndrome of . Syndromes uniquely label cosets in the quotient.

• In a given coset a minimum weight string is called a coset leader, and the correction of a string is . Thus a perfect code has unique coset leaders.

• Linear equivalence of codes

Properties

• The ^rate of a -code is .
• The ^minimumDistance of a linear code is its ^minimumWeight.

Special kinds of linear code

• Projective code

See also

• Extended code

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Footnotes

1. 1999. Introduction to coding theory, §3.2, pp. 35–36 ↩