On a necessary condition for L-step orthogonalization of linear codes and its applications

By generalizing a known result for double-error-correcting binary primitive Bose\3-Chaudhuri\3-Hocquenghem codes, the following statement is proved: A necessary condition for L-step orthogonalization of a t-error-correcting binary linear code C of length n is given by n ⩾ {if123-1} + {if123-2} if its expurgated version C′ is distinct from C, and n ⩾ {if123-3} if C′ is identical to C, where {if123-4} is the minimum distance of the dual {if123-5} of the code C′, and {if123-6} is that of the dual {if123-7} of the code C′, expurgated from C by a single information digit.From this result, it is found that the following codes cannot be L-step orthogonalized: triple-error-correcting BCH codes with odd m greater than 5 and even m equal to 8 and 10, all the BCH codes with m = 7 except two trivial cases, and most of the binary quadratic residue codes with known minimum Hamming distances. In addition, various codes are analyzed; and where possible, the types of majority-logic decoding are tabulated for these codes, namely, a class of binary cyclic codes of length n ⩽ 43 and all the binary BCH codes of length n = 63