Lexicodes over finite principal ideal rings

2018 ◽  
Vol 86 (11) ◽  
pp. 2661-2676
Author(s):  
Jared Antrobus ◽  
Heide Gluesing-Luerssen
Keyword(s):  
Principal Ideal ◽  
Finite Principal Ideal Rings

scholarly journals MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

2014 ◽  
Vol 125 ◽  
pp. 177-193 ◽  
Author(s):  
Marcus Greferath ◽  
Thomas Honold ◽  
Cathy Mc Fadden ◽  
Jay A. Wood ◽  
Jens Zumbrägel
Keyword(s):  
Principal Ideal ◽  
Extension Theorem ◽  
Macwilliams Extension Theorem ◽  
Finite Principal Ideal Rings

Finite Principal Ideal Rings

1976 ◽  
Vol 19 (3) ◽  
pp. 277-283 ◽  
Author(s):  
James L. Fisher
Keyword(s):  
Principal Ideal ◽  
General Setting ◽  
Finite Rings ◽  
Structure Theorems ◽  
Finite Principal Ideal Rings

This paper determines the structure of finite rings whose two sided ideals are principal as left ideals, and as right ideals. Such rings will be called principal ideal rings. Although finite rings have been studied extensively [1], [5], [12], [14] and the tools necessary for describing finite principal ideal rings have been available for thirty years, these structure theorems (which are essentially given in a more general setting in [4]) seem to have been overlooked. In particular, let or be an endomorphism of a ring V.

scholarly journals Determinants of some special matrices over commutative finite chain rings

2020 ◽  
Vol 8 (1) ◽  
pp. 242-256
Author(s):  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Principal Ideal ◽  
Residue Field ◽  
Circulant Matrices ◽  
Finite Chain ◽  
Algebraic Structures ◽  
Open Problems ◽  
Finite Chain Rings ◽  
Positive Integers ◽  
Finite Principal Ideal Rings

AbstractCirculant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

MDS codes over finite principal ideal rings

2008 ◽  
Vol 50 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Steven T. Dougherty ◽  
Jon-Lark Kim ◽  
Hamid Kulosman
Keyword(s):  
Principal Ideal ◽  
Mds Codes ◽  
Finite Principal Ideal Rings
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