The celebrated Feng-Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. The new bound is very much related to the Feng-Rao bound as well as to Shibuya and Sakaniwa's bound in [T. Shibuya, K. Sakaniwa, A dual of well-behaving type designed minimum distance, IEICE Trans. Fund. E84-A (2001) 647-652]. Our bound is easily extended to deal with any generalized Hamming weights. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains.

The Feng-Rao bound gives a lower bound on the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. Actually our result not only deals with the minimum distance but gives lower bounds on any generalized Hamming weight. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains. The improved codes from the present paper are not in general equal to the Feng-Rao improved codes but the constructions are very much related.

Constructing new codes from existing ones by puncturing is in this paper viewed in the context of order domains R where puncturing can be seen as redefinition of the evaluation map : R → F n q . The order domains considered here are of the form R =F[x 1 , x 2 , . . . , x m ]/I where redefining can be done by adding one or more polynomials to the basis of the defining ideal I to form a new ideal J in such a way that the number of points in the variety V(I ) is reduced by t to form V(J ) and puncturing in t coordinates is achieved. An explicit construction of such polynomials is given in the case of codes defined by Norm-Trace curves and examples are given of both evaluation codes and dual codes. Finally, it is demonstrated that the improvement in minimum distance can be significant when compared to the lower bound obtained by ordinary puncturing.