Definitions

# Justesen code

In coding theory, Justesen codes form a class of error-correcting codes which are derived from Reed-Solomon codes and have good error-control properties.

## Definition

Let R be a Reed-Solomon code of length N = 2m − 1, rank K and minimum weight N − K + 1. The symbols of R are elements of F = GF(2m) and the codewords are obtained by taking every polynomial ƒ over F of degree less than K and listing the values of ƒ on the non-zero elements of F in some predetermined order. Let α be a primitive element of F. For a codeword a = (a1, ..., aN) from R, let b be the vector of length 2N over F given by

$mathbf\left\{b\right\} = left\left(a_1, a_1, a_2, alpha^1 a_2, ldots, a_N, alpha^\left\{N-1\right\} a_N right\right)$

and let c be the vector of length 2N m obtained from b by expressing each element of F as a binary vector of length m. The Justesen code is the linear code containing all such c.

## Properties

The parameters of this code are length 2m N, dimension m K and minimum distance at least

$sum_\left\{i=1\right\}^ell i binom\left\{2m\right\}\left\{i\right\} .$

The Justesen codes are examples of concatenated codes.

## References

• J. Justesen (1972). "A class of constructive asymptotically good algebraic codes". IEEE Trans. Info. Theory 18 652–656.
• F.J. MacWilliams; N.J.A. Sloane The Theory of Error-Correcting Codes. North-Holland.

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