Cyclotomic units and Stickelberger ideals of global function fields

In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.

     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

Self-Orthogonal Greedy Codes

In this paper, we examine greedily generated self-orthogonal codes. Weshow that they are linear, and discuss several properties whichthey possess. We also look at specific examples of these codes.     

Almost MDS Codes

MDS codes are codes meeting the Singleton bound. Both for theory and practice, these codes are very important and have been studied extensively. Codes near this bound, but not attaining it, have had far less attention. In this paper we study codes that almost reach the Singleton bound.     

Co-Orthogonal Codes

We define, construct and sketch possible applications of a new class of non-linear codes: co-orthogonal codes, with possible applications in cryptography and parallel processing. We also describe a fast and general method for generating (non-linear) codes with prescribed dot-products with the help of multi-linear polynomials.     

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.     

The Theory of Cyclic Codes and a Generalization to Additive Codes

We present a new approach to the theory of cyclic and constacyclic codes and generalize the theory to cover the family of additive (not necessarily linear) cyclic codes. The approach is based on the action of the Galois group (cyclotomic cosets). The conventional representation of cyclic codes as ideals in a factor ring of the polynomial ring is not needed.     

An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear [n,k,d] code, whose minimum distance is odd, can be extended to an [n + 1, k, d + 1] code. This is one of the few elementary results about binary codes which does not obviously generalise to q-ary codes. The aim of this paper is to give a simple sufficient condition for a q-ary [n, k, d] code to be extendable to an [n + 1, k, d + 1] code. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.     

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.     

Weight Enumerators of Double Circulant Codes and New Extremal Self-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.     

Structure Theorems for Group Ring Codes with an Application to Self-Dual Codes

Using ideas from the cohomology of finite groups, an isomorphism is established between a group ring and the direct sum of twisted group rings. This gives a decomposition of a group ring code into twisted group ring codes. In the abelian case the twisted group ring codes are (multi-dimensional) constacyclic codes. We use the decomposition to prove that, with respect to the Euclidean inner product, there are no self-dual group ring codes when the group is the direct product of a 2-group and a group of odd order, and the ring is a field of odd characteristic or a certain modular ring. In particular, there are no self-dual abelian codes over the rings indicated. Extensions of these results to non-Euclidean inner products are briefly discussed.     

Asymptotically Good Codes from Generalized Algebraic-Geometry Codes

We consider generalized algebraic-geometry codes, based on places of the same degree of a fixed algebraic function field over a finite field. In this note, using a method similar to the Justesen’s one, we construct a family of such codes which is asymptotically good.Communicated by: D. Jangnickel     

A Class of Perfect Ternary Constant-Weight Codes

We construct a class of perfect ternary constant-weight codes of length 2 ^r , weight 2 ^r -1 and minimum distance 3. The codes have codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes.     

Error-Correcting Codes over an Alphabet of Four Elements

The problem of finding the values of A_q(n,d)—the maximum size of a code of length n and minimum distance d over an alphabet of q elements—is considered. Upper and lower bounds on A₄(n,d) are presented and some values of this function are settled. A table of best known bounds on A₄(n,d) is given for n 12. When q M < 2q, all parameters for which A_q(n,d) = M are determined.     

有限域上的重根自对偶负循环码

Let F_q be a finite field with q = p~m, where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2~ap~r over F_q, a ≥ 1.The construction of self-dual negacyclic codes of length 2~abp~r over F_q is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.