The minimum distance of all binary cyclic codes of odd lengths from 69 to 99

A computer search has been made to determine the true minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> for all binary cyclic codes having odd lengths <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> in the range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">69≤ n≤ 99</tex> . Using an algorithm originally developed by C. L. Chen, the generator matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> of each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k)</tex> binary cyclic code was put in systematic form. All possible codewords obtained from sums of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> rows of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> , for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i= 1,2, ⋯,υ</tex> , were examined, and the minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dυ</tex> of this set was recorded. Then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d=dυ</tex> whenever <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">υ >\{(dυ-l)k/n\}-l</tex> . Known equivalences among cyclic codes were taken into account, and only one code from each equivalence class was listed. Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x)</tex> divide <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xⁿ-1</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x - 1</tex> is not a factor of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x)</tex> . Then the minimum distances of the codes generated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x),(x - l)g(x)</tex> and their duals are listed together. For each such set of codes, the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">υ</tex> for which a codeword of minimum weight first appeared is listed. The codes found were compared with the list of best codes tabulated by Sloane [5]. Many good cyclic codes have been found. Among the best <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k,d)</tex> cyclic codes found are the following: (73, 27, 20), (73, 36, 16), (85, 12, 34), (85, 20, 28), (87, 31, 22), (89, 56, 11), (91, 51, 14), (93, 20, 32), (93, 23, 29), (93, 31, 24), (93, 33, 22).