scholarly journals Lee weights of cyclic self-dual codes over Galois rings of characteristic p 2

2017 ◽  
Vol 45 ◽  
pp. 107-130 ◽  
Author(s):  
Boran Kim ◽  
Yoonjin Lee
2019 ◽  
Vol 12 (4) ◽  
pp. 1701-1716
Author(s):  
Trilbe Lizann Espina Vasquez ◽  
Gaudencio Jr. Cempron Petalcorin

Let $p$ be an odd prime and $r$ a positive integer. Let $\text{GR}(p^3,r)$ be the Galois ring of characteristic $p^3$ and cardinality $p^{3r}$. In this paper, we investigate the self-dual codes over $\text{GR}(p^3,r)$ and give a method to construct self-dual codes over this ring. We establish a mass formula for self-dual codes over $\text{GR}(p^3,r)$ and classify self-dual codes over $\text{GR}(p^3,2)$ of length 4 for $p=3,5$.


2007 ◽  
Vol 45 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Jon-Lark Kim ◽  
Yoonjin Lee
Keyword(s):  

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 303
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.


Information ◽  
2019 ◽  
Vol 10 (4) ◽  
pp. 153
Author(s):  
Sunghyu Han

In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings G R ( p m , r ) with p ≡ − 1 ( mod 4 ) and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over G R ( p m , 3 ) with ( p = 3 and m = 2 , 3 , 4 , 5 , 6 ), ( p = 7 and m = 2 , 3 ), ( p = 11 and m = 2 ), ( p = 19 and m = 2 ), ( p = 23 and m = 2 ), and ( p = 31 and m = 2 ). In the building-up construction, it is important to determine the existence of a square matrix U such that U U T = − I , which is called an antiorthogonal matrix. We prove that there is no 2 × 2 antiorthogonal matrix over G R ( 2 m , r ) with m ≥ 2 and odd r.


2006 ◽  
Vol 02 (02) ◽  
pp. 289-303 ◽  
Author(s):  
PHILIPPE GABORIT ◽  
ANN MARIE NATIVIDAD ◽  
PATRICK SOLÉ

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.


2004 ◽  
Vol 03 (01) ◽  
pp. 31-48 ◽  
Author(s):  
S. GONZÁLEZ ◽  
C. MARTÍNEZ ◽  
I. F. RÚA ◽  
V. T. MARKOV ◽  
A. A. NECHAEV

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.


2020 ◽  
Vol 54 (1-3) ◽  
pp. 198-228
Author(s):  
Gary Marker

Abstract This essay constitutes a close reading of the works of Feofan Prokopovich that touch upon gender and womanhood. Interpretively it is informed by Judith Butler’s book Gender Trouble, specifically by her model of gender-as-performance. Prokopovich’s writings conveyed a negative characterization of holy women and Russian women of power, a combination of glaring silences and Scholastic dual codes that in toto denied the association of womanhood with glory or wisdom. In this he stood apart from other East Slavic Orthodox homilists of his day, even though they too invariably associated virtue with masculinity (muzhestvo). For Prokopovich, wisdom, strength, constancy, etc., were innately masculine. Women, by contrast, were weak, inconstant, non-rational, and guided by emotion. His sermons nominally in praise of Catherine I and Anna Ioannovna were suffused with narrative gestures that, to those attuned to the nuances of Scholastic rhetoric, ran entirely counter to their nominal message. Several panegyrics to Anna, for example, made no mention of her at all, a practice in sharp contrast to his sermons to male rulers, which typically placed the honoree firmly in the foreground. Even more startling is his singularly minimalist approach to Mary, for whom he composed almost no sermons and whose presence he barely mentioned in tracts where one would have expected otherwise. This essay concludes that this attitude reflected both his personal preferences and influence that Protestant Pietism had on his thinking.


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