A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

A note on one-point quadrature formula for Daubechies scale function with partial support

This note is concerned with an efficient computation of integrals of products of a smooth function and Daubechies scale function with partial support by using a one-point quadrature rule. The error estimate is obtained. The rule is illustrated by considering an example from the literature.     

Dense packings from quadratic fields and codes

In the present paper, we make use of the quadratic field to construct dense packings in the Euclidean spaces. With the help from good error-correcting codes, we are able to produce several packings with the best-known densities. Furthermore, if we assume that the best upper bound in coding theory developed by Aaltonen, Ben-Haim and Litsyn could be achieved, then the Minkowski bound would be improved.     

A tower with non-Galois steps which attains the Drinfeld-Vladut bound

We introduce a new tower of function fields over a finite field of square cardinality, which attains the Drinfeld-Vladut bound. One new feature of this new tower is that it is constructed with non-Galois steps; i.e., with non-Galois function field extensions. The exact value of the genus g(F_n) is also given (see Lemma 4).     

A simplified proof for the limit of a tower over a cubic finite field

Recently Bezerra, Garcia and Stichtenoth constructed an explicit tower F=(Fn)_n?0 of function fields over a finite field Fq³, whose limit λ(F)=limn→∞N(Fn)/g(Fn) attains the Zink bound λ(F)?2(q²−1)/(q+2). Their proof is rather long and very technical. In this paper we replace the complex calculations in their work by structural arguments, thus giving a much simpler and shorter proof for the limit of the Bezerra, Garcia and Stichtenoth tower.     

Further improvements on the Feng–Rao bound for dual codes

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.     

Asymptotically good towers of function fields with small p-rank

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

On towers of function fields of Artin-Schreier type

In this article we derive strong conditions on the defining equations of asymptotically good Artin-Schreier towers. We will show that at most three kinds of defining equations can give rise to a recursively defined good tower, if we restrict ourselves to prime degrees. ¹A. Garcia and H. Stichtenoth did part of thiswork during their stay at Sabanci University, Istanbul, Turkey (Sept. 2002). ²A. Garcia was partially supported by PRONEX # 662408/1996-3 (CNPq-Brazil).     

A method for efficiently computing the number of codewords of fixed weights in linear codes

The problem of computing the number of codewords of weights not exceeding a given integer in linear codes over a finite field is considered. An efficient method for solving this problem is proposed and discussed in detail. It builds and uses a sequence of different generator matrices, as many as possible, so that the identity matrix takes disjoint places in them. The efficiency of the method is achieved by optimizations in three main directions: (1) the number of the generated codewords, (2) the check whether a given codeword is generated more than once, and (3) the operations for generating and computing these codewords. Since the considered problem generalizes the well-known problems “Weight Distribution” and “Minimum Distance”, their efficient solutions are considered as applications of the algorithms from the method.     

Witt equivalence of function fields of curves over local fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13 G?adki, P., Marshall, M. Witt equivalence of function fields over global fields. Trans. Am. Math. Soc., electronically published on April 11, 2017, doi: https://doi.org/10.1090/tran/6898 (to appear in print).[Crossref] , [Google Scholar]] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.     

Distance regularity of Kerdock codes

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.     

The covering radius of extreme binary 2-surjective codes

The covering radius of binary 2-surjective codes of maximum length is studied in the paper. It is shown that any binary 2-surjective code of M codewords and of length has covering radius if M − 1 is a power of 2, otherwise . Two different combinatorial proofs of this assertion were found by the author. The first proof, which is written in the paper, is based on an existence theorem for k-uniform hypergraphs where the degrees of its vertices are limited by a given upper bound. The second proof, which is omitted for the sake of conciseness, is based on Baranyai’s theorem on l-factorization of a complete k-uniform hypergraph.      

Graphs and recursively defined towers of function fields

In this paper we state and explore a connection between graph theory and the theory of recursively defined towers. This leads, among other things, to a generalization of Lenstra's identity (Finite Fields Appl. 8 (2001) 166) and the solution of an open problem concerning the Deuring polynomial posed in (J. Reine Angew. Math. 557 (2003) 53). Further we investigate the effect extension of the constant field has on the limit of certain towers.