Olav Geil

Evaluation codes from order domain theory

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 Sakaniwas 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.

On the order bounds for one-point AG codes

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [H. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields and their Applications 14 (2008), pp. 92-123]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound

On Hyperbolic Codes

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Footprints or generalized Bezout's theorem

for the minimum distance of these codes. We establish a connection between

Relative generalized Hamming weights of one-point algebraic geometric codes

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Feng-Rao decoding of primary codes

and the order bound and its generalizations. We also study the improved code constructions based on

Bounding the number of Fq-rational places in algebraic function fields using Weierstrass semigroups

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On the feng-rao bound for generalized hamming weights

. Finally we extend

An improvement of the Feng---Rao bound for primary codes

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List decoding algorithm based on voting in Gröbner bases for general one-point AG codes

to all generalized Hamming weights.