THE MACWILLIAMS-SLOANE CONJECTURE ON THE TIGHTNESS OF THE CARLITZ-UCHIYAMA BOUND AND THE WEIGHTS OF DUALS OF BCH CODES

Research Problem 9.5 of MacWilliams and Sloane's book, The Theory of E rror Correcting Codes (Amsterdam: North-Holland, 1977), asks for an im provement of the minimum distance bound of the duals of BCH codes, def ined over F2m with m odd. The objective of the present article is to g ive a solution to the above problem by with (i) obtaining an improveme nt to the Ax theorem, which we prove is the best possible for many cla sses of examples; (ii) establishing a sharp estimate for the relevant exponential sums, which implies a very good improvement for the minimu m distance bounds; (iii) providing a doubly infinite family of counter examples to Problem 9.5 where both the designed distance and the lengt h increase independently; (iv) verifying that our bound is tight for s ome of the counterexamples; and (v) in the case of even m, giving a do ubly infinite family of examples where the Carlitz-Uchiyama bound is t ight, and in this way determining the exact minimum distance of the du als of the corresponding BCH codes.