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 F/sub 2/m 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.>

For pt.I, see Proc. AMS, vol.III, p.523-31 (1991). The minimum distance of a Goppa code is found when the length of code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri (1966). This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in other words, the covering radii are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code is found. Finally, a bound on the minimum distance of the dual code is discussed.>