Geodesic
Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo $2$
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2014), pp. 62-71 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, vector spaces over residue field modulo $2$ are considered. These vector spaces are of considerable interest because they are widely used in ordinary graphs theory, theory of coding and others areas, especially in modular systems investigation. Vector spaces over $\mathrm{GF(2)}$ have some features, for example, examination of linear dependence and independence for the set of vectors is simplified. The concept of $1$-dependence for the set of vectors is embedded. This concept is used to study vector subspaces and their orthogonal complements and to solve systems of linear equations. The connection between fundamental system of solutions of some simultaneous linear equations and decomposition of corresponding vector system into minimal $1$-dependent subsystems is considered. The necessary and sufficient conditions for the existence of nontrivial intersection of the vector subspace and its orthogonal complement are proven. Bibliogr. 10.
Keywords: vector space, residue field modulo 2. 
@article{VSPUI_2014_1_a6,
     author = {E. A. Kalinina and G. M. Khitrov},
     title = {Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo~$2$},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {62--71},
     year = {2014},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a6/}
}
TY  - JOUR
AU  - E. A. Kalinina
AU  - G. M. Khitrov
TI  - Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo $2$
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2014
SP  - 62
EP  - 71
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a6/
LA  - ru
ID  - VSPUI_2014_1_a6
ER  - 
%0 Journal Article
%A E. A. Kalinina
%A G. M. Khitrov
%T Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo $2$
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2014
%P 62-71
%N 1
%U http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a6/
%G ru
%F VSPUI_2014_1_a6
E. A. Kalinina; G. M. Khitrov. Peculiar properties of vector space of ordered $(0,1)$ $n$-tuples of elements over residue field modulo $2$. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2014), pp. 62-71. http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a6/

[1] Swamy M. N. S., Thulasiraman K., Graphs: Theory and Algorithms, Per. s angl., Mir, M., 1984, 454 pp.

[2] Deo N., Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, New York, 1974, 478 pp. | MR | Zbl

[3] Tarakanov V. E., Combinatorial problems and $(0,1)$-matrices, Nauka, M., 1985, 192 pp. | MR

[4] Seelinger G., Sissokho P., Spence L., Vanden Eynden C., “Partitions of finite vector spaces over $GF(2)$ into subspaces of dimensions 2 and $s$”, Finite Fields and Their Applications, 18:6 (2012), 1114–1132 | DOI | MR | Zbl

[5] Coppersmith D., “Solving linear equations over $GF(2)$: block Lanczos algorithm”, Linear Algebra and its Applications, 192 (1993), 33–60 ; 36:3–4 (1929), 205–338 | DOI | MR | Zbl

[6] Gegalkin I. I., “L'arithmétisation de la logique symbolique”, Mat. sb., 35:3–4 (1928), 311–377 ; 36:3–4 (1929), 205–338

[7] Lidl R., Niederreiter H., Finite fields, in 2 vol. Per. s angl., v. 1, Mir, M., 1988, 430 pp. | MR | Zbl

[8] Couceiro M., Foldes S., “Definability of Boolean function classes by linear equations over $GF(2)$”, Discrete Applied Mathematics, 142:1–3 (2004), 29–34 | DOI | MR | Zbl

[9] Faddeev D. K., Lectures on algebra. Textbook, Nauka, M., 1984, 416 pp. | MR

[10] Babai L., Frankl P., Linear Algebra Methods in Combinatorics with Applications to Geometry and Computer Science, Preliminary version 2 (September 1992), The University of Chicago, Chicago, 1992, 216 pp. | Zbl