scholarly journals Matrix of ℤ-module

2015 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Linear Transformation ◽  
Bilinear Form ◽  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Gramian Matrix ◽  
Selection Of

Summary In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

scholarly journals Torsion Z-module and Torsion-free Z-module

2014 ◽  
Vol 22 (4) ◽  
pp. 277-289
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Finitely Generated ◽  
Torsion Free

Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

scholarly journals Free ℤ-module

2012 ◽  
Vol 20 (4) ◽  
pp. 275-280
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Reduction Algorithm ◽  
Selection Of

Summary In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

CODING THEORY ON THE (m, t)-EXTENSION OF THE FIBONACCI p-NUMBERS

2011 ◽  
Vol 03 (02) ◽  
pp. 259-267 ◽  
Author(s):  
MANJUSRI BASU ◽  
BANDHU PRASAD
Keyword(s):  
Coding Theory ◽  
Negative Integer ◽  
Matrix Elements ◽  
Proper Selection ◽  
Code Matrix ◽  
Selection Of

In this paper, we define (m, t)-extension of the Fibonacci p-numbers and Golden (p, m, t)-proportions, where p ≥ 0 is integer, m > 0, and t > 0. We establish a relation among golden (p, m, t)-proportion, golden (p, m)-proportion and golden p-proportion. Thereby, we define a new Fibonacci Gp, m, t matrix. Then we show that by proper selection of the initial terms for the (m, t)-extension of the Fibonacci p-numbers, we can apply Fibonacci coding/decoding in Gp, m, t matrix. Also it is obvious that for t = 1, the relations among the code elements for all values of p (non-negative integer) and m (> 0) coincide with the relations among the code matrix elements for all values of p and m (> 0) with the same initial terms (see the paper coding theory on the m-extension of the Fibonacci p-numbers, Chaos, Solitons and Fractals42 (2009) 2522–2530).

scholarly journals Transitivity in integral symplectic forms

1968 ◽  
Vol 8 (1) ◽  
pp. 43-48 ◽  
Author(s):  
D. G. James
Keyword(s):  
Bilinear Form ◽  
Finite Rank ◽  
Bilinear Mapping ◽  
Symplectic Forms

A Symplectic lattice L is a free Z-module of finite rank endowed with a non-degenerate alternating bilinear form. Thus we have a bilinear mapping Φ of L × L into the domain of integers Z; we donote Φ (α, β) by α · β (where α, β ∈ L). Then α2 = 0 and α·β = −β·α.

scholarly journals Divisible ℤ-modules

2016 ◽  
Vol 24 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Coefficient Ring ◽  
Torsion Free

Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

scholarly journals Lattice of ℤ-module

2016 ◽  
Vol 24 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Coding Theory ◽  
Positive Definite ◽  
Bilinear Forms ◽  
Reduction Algorithm ◽  
Real Numbers ◽  
Scalar Products ◽  
Definition Of

Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

scholarly journals Integrated On-Line and Off-Line Error Detection Mechanisms in the Coding Theory Framework

VLSI Design ◽  
1998 ◽  
Vol 5 (4) ◽  
pp. 313-331
Author(s):  
Mark G. Karpovsky
Keyword(s):  
Error Detection ◽  
Coding Theory ◽  
Space Time ◽  
Time Compression ◽  
Manufacturing Testing ◽  
Aliasing Probability ◽  
On Line ◽  
Detection Mechanisms ◽  
Theory Framework ◽  
Selection Of

In this paper we present an approach for combining on-line concurrent checking (CC) with off-line built-in self-test (BIST). We will show that a reduction of an aliasing probability can be obtained for manufacturing testing by monitoring the output of a concurrent checker and a reduction of a probability of not detecting an error in the computing mode can be obtained by a short periodic BIST. We will present a technique for optimal selection of error-detecting codes for combined on-line CC and off-line space-time compression of test responses for BIST and estimate probabilities of not detecting an error for the approach based on integrating CC and BIST. We also present a technique for on-line error-detection in space-time compressors of test responses for BIST.

Sign in / Sign up

Export Citation Format

Share Document