The True Minimum Distance of Some Narrow-Sense BCH-Codes of Length 255

Using equivalent codes it is shown that the BCH-bound for the following narrowsense BCH-codes already yields the true minimum distance: [255,87,53], [255,107,45], [255,115,43], [255,123,39], [255,131,37], [255,147,29], [255,163,25], [255,179,21]. For the remaining two narrow-sense BCH-codes of length 255 in the book of F. J. MacWilliams and N. J. A. Sloane, page 261, Figure 9.1, whose true minimum distances are still unknown, upper bounds on the minimum distances are given which differ only by two from the corresponding BCH-bounds. Even 16 years after the first printing of "The Theory of Error-Correcting Codes" the table of primitive BCH-codes of length up to 255 ([3], 7 th printing, p.261) contains ten entries where the true minimum distances are unknown. We show that the BCHbound gives the true minimum distance for the following codes: [255,87,53], [255,107,45], [255,115,43], [255,123,39], [255,131,37], [255,147,29], [255,163,25], [255,179,21] by explicitly giving a check polynomia..