A new class of recurrent codes

Abstract

Recurrent codes for burst-error correction are defined to be a type B1 code. A type B1 code is capable of correcting all the single burst errors of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> or less. A type B2 code is a subclass of a type B1 code with the restriction that the single burst error of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> or less occurs within <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> consecutive blocks (it is assumed that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> is divisible by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l = rb</tex> ). In this paper, the author will present a class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -nary recurrent codes--both type B1 and B2 codes--for burst-error correction. The construction procedures for this class of codes are simple and systematic. An interesting relation exists between a type B1 and a type B2 code. If a code is type B2 (with block length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> , burst-error-correction capability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l = rb</tex> ), then by reducing the block length by one symbol, the type B2 code becomes a type B1 code, while the burst-error-correction capability remains the same. Both types of codes can be used for the correction of binary burst errors. The correction of binary burst errors will be briefly discussed in this paper.