On the minimum distance of some quadratic residue codes (Corresp.)

Abstract

We find the minimum distances of the binary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(113, 57)</tex> , and ternary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(37, 19), (61, 31), (71, 36)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(73, 37)</tex> quadratic residue codes and the corresponding extended codes. These distances are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">15, 10, 11, 17</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">17</tex> , respectively, for the nonextended codes and are increased by one for the respective extended codes. We also characterize the minimum weight codewords for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(113, 57)</tex> binary code and its extended counterpart.