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" = 2^"m" − 1, rank "K" and minimum weight "N" − "K" + 1. The symbols of "R" are elements of "F" = GF(2^"m") 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 = ("a"₁, ..., "a"_"N") from "R", let b be the vector of length 2"N" over "F" given by

:$mathbf\{b\}\; =\; left(\; a\_1,\; a\_1,\; a\_2,\; alpha^1\; a\_2,\; ldots,\; a\_N,\; alpha^\{N-1\}\; a\_N\; ight)$

and let c be the vector of length 2"N" "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 2"m" "N", dimension "m" "K" and minimum distance at least

:

The Justesen codes are examples of concatenated codes.

References

*
*