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Kopparty and Wang studied in [3] the relation between the roots of a univariate polynomial over F q and the zero–nonzero pattern of its coefficients. We generalize their results to polynomials in more variables. [ABSTRACT FROM AUTHOR]

Salazar, Dunn and Graham in [16] presented an improved Feng-Rao bound for the minimum distance of dual codes. In this work we take the improvement a step further. Both the original bound by Salazar et al., as well as our improvement are lifted so that they deal with generalized Hamming weights. We also demonstrate the advantage of working with one-way well-behaving pairs rather than weakly well-behaving or well-behaving pairs. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Feng–Rao bound for dual codes and a similar bound by Andersen and Geil (2008) [1] for primary codes are consequences of each other. This implies that the Feng–Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura (2000) [30] (see also Beelen and Høholdt, 2008 [3]) derived from the Feng–Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition in Matsumoto and Miura (2000) [30] requires the use of differentials which was not needed in Andersen and Geil (2008) [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miuraʼs bound and Andersen and Geilʼs bound when applied to primary one-point algebraic geometric codes. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

The main results of this paper are derived by using only simple Gro¨bner basis techniques. We present a new construction of evaluation codes from Miura–Kamiya curves Cab. We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometric Goppa codes. In particular we consider codes from norm–trace curves. We show that our estimates give actually the true minimum distance of these codes. The new codes from norm–trace curves perform rather well. In many cases much better than the corresponding geometric Goppa codes. It turns out that an alternative description of the new codes from norm–trace curves can be made by using Høholdt et al.'s in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, Vol. 1, Elsevier, Amsterdam, 1998, pp. 871–961 (Chapter 10) construction of improved dual codes. [Copyright &y& Elsevier]

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gro¨bner algebras, and graded structures is given. The theory of Gro¨bner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem, hold. The dimension of an order domain is related to the rank of its value semigroup. [Copyright &y& Elsevier]

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [15] and in the CSS construction of quantum codes [14]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance. [ABSTRACT FROM AUTHOR]