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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]

1996
*
Lecture Notes in Computer Science
*

Their parameters are derived from bounds

doi:10.1007/3-540-68697-5_3
fatcat:q7fpbgmcfvevndenxmodnujcrm
*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. ...##
###
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*. ...

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

2016
*
arXiv
*
pre-print

We proof that

arXiv:1602.01242v2
fatcat:qpw3im4hufeyphpinvnfdxhz4m
*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*. ...##
###
Abstract algebra, projective geometry and time encoding of quantum information
[article]

2005
*
arXiv
*
pre-print

Some

arXiv:quant-ph/0503159v1
fatcat:6kvgqck33zashjpxoezc56uwtm
*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*. ...##
###
p-Adic estimates of hamming weights in abelian codes over galois rings

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*. ...

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

2011
*
IEEE Transactions on Information Theory
*

the Teichmüller set of and Tr(1) is the

doi:10.1109/tit.2011.2156382
fatcat:ldiraz5kvjcttkbjzpgvh7523a
*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). ...##
###
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)

2004
*
European journal of combinatorics (Print)
*

The invariant polynomials can be constructed from the weight enumerators of type II

doi:10.1016/j.ejc.2003.01.001
fatcat:cmmbyhdwkfe6jijhyk6x4g2r6a
*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*. ...##
###
Additive cyclic codes over finite commutative chain rings
[article]

2017
*
arXiv
*
pre-print

Additive cyclic

arXiv:1701.06672v1
fatcat:7sm44ll55valpm7xxv3kqphsey
*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). ...##
###
Self-Dual Normal Basis of a Galois Ring

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] . ...

##
###
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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