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Matrix of ℤ-module1

100%

Futa Y., Okazaki H., Shidama Y.

Formalized Mathematics

2015

tom 23

nr 1

29-49

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.

Feedback equivalence of convolutional codes over finite rings

84%

DeCastro-García N.

Open Mathematics

2017

tom 15

nr 1

1495-1508

The approach to convolutional codes from the linear systems point of view provides us with effective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.

Ograniczanie wyników

1 Formalized Mathematics

1 Open Mathematics

1 DeCastro-García N.

1 Futa Y.

1 Okazaki H.

1 Shidama Y.

1 2017

1 2015

Wyniki wyszukiwania

Matrix of ℤ-module1

Feedback equivalence of convolutional codes over finite rings