Short binary convolutional codes with maximal free distance for rates 2/3 and 3/4 (Corresp.)

Abstract

A search procedure is developed to find good short binary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N,N - 1)</tex> convolutional codes. It uses simple rules to discard from the complete ensemble of codes a large fraction whose free distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{free}</tex> either cannot achieve the maximum value or is equal to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{free}</tex> of some code in the remaining set. Farther, the search among the remaining codes is started in a subset in which we expect the possibility of finding codes with large values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{free}</tex> to be good. A number of short, optimum (in the sense of maximizing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{free}</tex> ), rate-2/3 and 3/4 codes found by the search procedure are listed.