利用有限域GF（2m）上的三项式构造二元循环码

循环码是一类特殊的线性码,由于循环码快速的编码和译码算法,它被广泛应用于消费电子,数据存储以及通信系统当中.在本文中,利用特征是偶数的有限域上的三项式构造出了两类二元循环码,我们不仅可以确定出这两类循环码最小距离的下界,而且这两类循环码在参数的选取上非常的灵活.     

Montgomery Multiplication in GF(2k)

We show that the multiplication operation c=a · b · r^-1 in the field GF(2^k can be implemented significantly faster in software than the standard multiplication, where r is a special fixed element of the field. This operation is the finite field analogue of the Montgomery multiplication for modular multiplication of integers. We give the bit-level and word-level algorithms for computing the product, perform a thorough performance analysis, and compare the algorithm to the standard multiplication algorithm in GF(2^k. The Montgomery multiplication can be used to obtain fast software implementations of the discrete exponentiation operation, and is particularly suitable for cryptographic applications where k is large.     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

Optimal curves of low genus over finite fields

We investigate maximal and minimal curves of genus 4 and 5 over finite fields with discriminant −11 and −19. As a result the Hasse–Weil–Serre bound is improved.     

A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems.     

The Weight Enumerator of Linear Codes over GF q m) Having Generator Matrix over GF q

We have the relationships between the Hamming weight enumerator of linear codes over GFq ^m which have generator matrices over GFq, the support weight enumerator and the -ply weight enumerator.     

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.     

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation : GF(q^s) GF(q^s) can be used to treat a linear cyclic code of wordlength v over GF(q^s) as a linear code () of Wordlength sv over GF(q). This paper determines those linear cyclic codes and transformations for which the resulting linear code () is also cyclic.     

有限域上最优正规基的乘法表

本文给出了有限域上最优正规基乘法表的一个计算方法,改进了孙琦的相应结果.在有限域上椭圆曲线密码体制的应用中,本文给出的算法是非常有效的.     

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

Bit-Parallel Arithmetic Implementations over Finite Fields GF(2 m ) with Reconfigurable Hardware

Galois (or finite) fields are used in a wide number of technical applications, playing an important role in several areas such as cryptographic schemes and algebraic codes, used in modern digital communication systems. Finite field arithmetic must be fast, due to the increasing performance needed by communication systems, so it might be necessary for the implementation of the modules performing arithmetic over Galois fields on semiconductor integrated circuits. Galois field multiplication is the most costly arithmetic operation and different approaches can be used. In this paper, the fundamentals of Galois fields are reviewed and multiplication of finite-field elements using three different representation bases are considered. These three multipliers have been implemented using a bit-parallel architecture over reconfigurable hardware and experimental results are presented to compare the performance of these multipliers.     

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn ². In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn ² points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).This research was partially supported by the National Science Foundation under Grants DMS-8521826 and DMS-8500494.     

GF（2）—modules for Sz（q）wrCt

LetVbeavectorspaceoverfieldK=GF(q),whereq=2",nodd.ThethereexistsinGL(V)somesubgroupH,whichisisomorphictoSz(q),andtheactionofHonVderivessomeirreduciblemodulestructureforH-LetF=GF(2),thenVcanbeseenasanFH-mod-ule,Vremainsirreducible,andVissaidtobeanaturalFH-module.Thenotionofnaturalmodule'forSz(q)canbegeneralizedtothewrethproductSz(q)wed(thedefinitionforthewreathproduct,seeLlj'9'l9),Thenotationsandterminol-ogyinthispaper'exceptforfewexceptions,areeitherstandardoeasiluaccessiblefromL2j'…     

Two New Infinite Families of 3-Designs from Kerdock Codes over Z4

In this paper we show that the support of the codewords of each type in the Kerdock code of length 2^m over Z₄ form 3-designs for any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer , whose parameters are ,and .     

On isogeny graphs of supersingular elliptic curves over finite fields

We study the isogeny graphs of supersingular elliptic curves over finite fields, with an emphasis on the vertices corresponding to elliptic curves of j-invariant 0 and 1728.     

Tangential 1-blocks over GF(3)

In a fundamental paper on the algebraic theory of graph colouring Tutte [2] proves that the only tangential 1-block over GF(2) is the geometry corresponding to the polygon matroid of the graph K₃. For a good discussion of the problem and a direct proof of Tutte's theorem see Aigner [1, p. 373].

The purpose of this note is to prove the following result.     

Combinatorial geometries representable over GF(3) and GF(q). I. The number of points

Letq be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rankn representable over GF(3) and GF(q) is at mostn ². Whenq is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2.This research was partially supported by National Science Foundation Grant DMS-8521826 and a North Texas State University Faculty Research Grant.