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Page 9004 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
(RS-MOSC-NIT; Moscow) Trace-function on a Galois ring in coding theory.  ...  Often they may be evaluated using numbers of some special solutions of the equation defined by the trace-function on a Galois ring.  ... 

Universal Hash Functions from Exponential Sums over Finite Fields and Galois Rings [chapter]

Tor Helleseth, Thomas Johansson
1996 Lecture Notes in Computer Science  
Their parameters are derived from bounds on exponential sums over finite fields and Galois rings. This is the first tirnr hash families based upon such exponential sums have 1)een considered.  ...  In t#liis 1)apcr ncw families of stmngly universal hash funct,ions, or equivalently, authentication codes, are proposed.  ...  We will next show that using the theory of exponential sums over Galois rings we can also construct universal hash families. 5 Some preliminaries on Galois rings are giveri below.  ... 
doi:10.1007/3-540-68697-5_3 fatcat:q7fpbgmcfvevndenxmodnujcrm

On the Hyper-Kloosterman Codes Over Galois Rings

2020 Journal of Mathematics Research  
In the present paper we define the Hyper-Kloosterman codes over Galois rings R(pe;m). We show that this code is the trace of linear code over R(pe;m).  ...  By the Hyper-Kloostermann sums over Galois rings, we determine the Hamming weight of any codeword of this code over Galois rings.  ...  , m) is a trace code over Galois ring.  ... 
doi:10.5539/jmr.v12n2p12 fatcat:6owfooxfovdxplw7lmuuune3uu

Galois Correspondence on Linear Codes over Finite Chain Rings [article]

A. Fotue Tabue and E. Martínez-Moro and C. Mouaha
2016 arXiv   pre-print
We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the  ...  Given S|R a finite Galois extension of finite chain rings and B an S-linear code we define two Galois operators, the closure operator and the interior operator.  ...  Restriction codes is a core topic in coding theory.  ... 
arXiv:1602.01242v2 fatcat:qpw3im4hufeyphpinvnfdxhz4m

Abstract algebra, projective geometry and time encoding of quantum information [article]

Michel R. P. Planat, Metod Saniga (ASTRONOMICAL Institute)
2005 arXiv   pre-print
Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.  ...  Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing.  ...  Three generic functions are met in elementary analytical number theory. One is the Mangoldt function, closely related to the above defined a(n) function.  ... 
arXiv:quant-ph/0503159v1 fatcat:6kvgqck33zashjpxoezc56uwtm

p-Adic estimates of hamming weights in abelian codes over galois rings

D.J. Katz
2006 IEEE Transactions on Information Theory  
In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more.  ...  A generalization of McEliece's theorem on the -adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with  ...  Now we work with a code over a Galois ring which is neither a finite field nor an integer residue ring.  ... 
doi:10.1109/tit.2005.864428 fatcat:3pd5om7wv5gybpxhjvf64e7myi

Several Classes of Codes and Sequences Derived From a $\BBZ_{4}$-Valued Quadratic Form

Nian Li, Xiaohu Tang, Tor Helleseth
2011 IEEE Transactions on Information Theory  
the Teichmüller set of and Tr(1) is the trace function from the Galois ring to 4 .  ...  Let m and k be positive integers with m=gcd(m;k) being odd, for a 2 and b 2 , the exponential sum x2 i T r(ax+2bx ) is studied systematically in this paper, where i = p 01, = (4; m) is a Galois ring, is  ...  A class of exponential sum over a Galois ring is denoted by where , is the Teichmüller set of and is the trace function from the Galois ring to (see Section II-B).  ... 
doi:10.1109/tit.2011.2156382 fatcat:ldiraz5kvjcttkbjzpgvh7523a

Page 2703 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
The element € is a root of a basic primitive polynomial, in a way a generator of the so-called Galois ring GR (4), which is a finite extension of the ring Z4.  ...  The basic facts about these Galois rings are given in the introduction. The dual of a quaternary code is defined in the usual way.  ... 

Jacobi forms over totally real fields and type II codes over Galois rings GR(2m,f)

K. Betsumiya, Y. Choie
2004 European journal of combinatorics (Print)  
The invariant polynomials can be constructed from the weight enumerators of type II codes over the Galois rings.  ...  In this paper, we study the invariant polynomial ring of the generalized Clifford-Weil group and give a connection to Jacobi modular forms over the totally real field.  ...  Introduction Recently there has been intensive research connecting invariant theory and coding theory over various rings.  ... 
doi:10.1016/j.ejc.2003.01.001 fatcat:cmmbyhdwkfe6jijhyk6x4g2r6a

Additive cyclic codes over finite commutative chain rings [article]

Edgar Martínez-Moro, Kamil Otal, Ferruh Özbudak
2017 arXiv   pre-print
Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings.  ...  One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes.  ...  Coding and Designs over GF(q).  ... 
arXiv:1701.06672v1 fatcat:7sm44ll55valpm7xxv3kqphsey

Self-Dual Normal Basis of a Galois Ring

Irwansyah, Intan Muchtadi-Alamsyah, Aleams Barra, Ahmad Muchlis
2014 Journal of Mathematics  
In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a  ...  LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings.  ...  On the other hand, as a generalization of Galois fields, Galois rings have also several connections with coding theory; see [5, 6] .  ... 
doi:10.1155/2014/258187 fatcat:r5fz6m2ojzhjlo2rhvdxosfbve

Page 696 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
A representation of an LRS over a Galois ring R in terms of the trace function from an extension ring S to R is derived, as well as a necessary and sufficient condition for an LRS to be a maximum-period  ...  Summary: “We study the polynomial representations (in one vari- able) of the Boolean functions on a Galois field of characteristic 2.  ... 

Page 5108 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
In this paper, a primitive sequence a over the Galois ring GR(4,r) and the first coordinate sequence a; of the sequence a are de- fined and the distributions of the elements in one period of the sequences  ...  First we obtain an estimate of 0,1 distribution in one period of a,_;. Then we give an estimate of the absolute value of the autocorrelation function |Cy(/)| of 3(2%¢ — 1)2"/? +.2°-!  ... 

Page 920 of Mathematical Reviews Vol. , Issue 84c [page]

1984 Mathematical Reviews  
Teruo Takeuchi (Niigata) 12C_Finite.fields, and finite commutative rings (number-theoretic aspects) See also 68031. Moreno, Oscar On primitive elements of trace equal to 1 in GF(2”').  ...  So, usual class field theory is interpreted as a theory on the relation between Fy K)( X) and nonramified abelian extensions of F in the case dim( X)=1.  ... 

Page 4711 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
4711 98g:94028 94A60 Helleseth, Tor (N-BERG-I; Bergen); Johansson, Thomas (S-LUND-IT; Lund) Universal hash functions from exponential sums over finite fields and Galois rings.  ...  Their parameters are derived from bounds on exponential sums over finite fields and Galois rings. This is the first time hash families based upon such exponential sums have been considered.  ... 
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