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 Error Correcting Codes (Amsterdam: North-Holland, 1977), asks for an improvement of the minimum distance bound of the duals of BCH codes, defined over F₂m with m odd. The objective of the present article is to give a solution to the above problem by with (i) obtaining an improvement to the Ax (1964) theorem, which we prove is the best possible for many classes of examples; (ii) establishing a sharp estimate for the relevant exponential sums, which implies a very good improvement for the minimum distance bounds; (iii) providing a doubly infinite family of counterexamples to Problem 9.5 where both the designed distance and the length increase independently; (iv) verifying that our bound is tight for some of the counterexamples; and (v) in the case of even m, giving a doubly infinite family of examples where the Carlitz-Uchiyama bound is tight, and in this way determining the exact minimum distance of the duals of the corresponding BCH codes