مرکز منطقه ای اطلاع رساني علوم و فناوري - The minimum distance of all binary cyclic codes of odd lengths from 69 to 99

Abstract :

A computer search has been made to determine the true minimum distance $d$ for all binary cyclic codes having odd lengths $n$ in the range $69\\leq n\\leq 99$ . Using an algorithm originally developed by C. L. Chen, the generator matrix $G$ of each $(n,k)$ binary cyclic code was put in systematic form. All possible codewords obtained from sums of $i$ rows of $G$ , for $i= 1,2, \\cdots ,\\upsilon$ , were examined, and the minimum distance $d_{\\upsilon }$ of this set was recorded. Then $d=d_{\\upsilon }$ whenever $\\upsilon > {(d_{\\upsilon }-l)k/n}-l$ . Known equivalences among cyclic codes were taken into account, and only one code from each equivalence class was listed. Let $g(x)$ divide $x^{n}-1$ , where $x - 1$ is not a factor of $g(x)$ . Then the minimum distances of the codes generated by $g(x),(x - l)g(x)$ and their duals are listed together. For each such set of codes, the value of $\\upsilon$ 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 $(n,k,d)$ 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).