scholarly journals Minimum weight words of binary codes associated with finite projective geometries

1985 ◽  
Vol 57 (3) ◽  
pp. 307-310 ◽  
Author(s):  
Bhaskar Bagchi ◽  
N.S. Narasimha Sastry
Keyword(s):  
Minimum Weight ◽  
Binary Codes ◽  
Finite Projective Geometries ◽  
Projective Geometries

Finite Projective Geometries

1952 ◽  
Vol 4 ◽  
pp. 302-313 ◽  
Author(s):  
Gerald Berman
Keyword(s):  
Explicit Construction ◽  
Primitive Elements ◽  
Galois Fields ◽  
Finite Projective Geometries ◽  
Linear Subspaces ◽  
Projective Geometries

James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, pn). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, pn). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.

New extremal Type II ℤ4-codes of length 32 obtained from Hadamard matrices

2019 ◽  
Vol 11 (05) ◽  
pp. 1950057
Author(s):  
Sara Ban ◽  
Dean Crnković ◽  
Matteo Mravić ◽  
Sanja Rukavina
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Hadamard Matrices ◽  
Binary Codes ◽  
Type Ii ◽  
Hadamard Design ◽  
Binary Linear Codes ◽  
Type Formula ◽  
Hadamard Designs

For every Hadamard design with parameters [Formula: see text]-[Formula: see text] having a skew-symmetric incidence matrix we give a construction of 54 Hadamard designs with parameters [Formula: see text]-[Formula: see text]. Moreover, for the case [Formula: see text] we construct doubly-even self-orthogonal binary linear codes from the corresponding Hadamard matrices of order 32. From these binary codes we construct five new extremal Type II [Formula: see text]-codes of length 32. The constructed codes are the first examples of extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text], [Formula: see text], whose residue codes have minimum weight 8. Further, correcting the results from the literature we construct 5147 extremal Type II [Formula: see text]-codes of length 32 and type [Formula: see text].

On Two Doubly Even Self-Dual Binary Codes of Length<tex>$160$</tex>and Minimum Weight<tex>$24$</tex>

2005 ◽  
Vol 51 (1) ◽  
pp. 408-410 ◽  
Author(s):  
M. vanDijk ◽  
S. Egner ◽  
M. Greferath ◽  
A. Wassermann
Keyword(s):  
Minimum Weight ◽  
Binary Codes

scholarly journals FINITE GEOMETRIES WITH QUBIT OPERATORS

2009 ◽  
Vol 12 (02) ◽  
pp. 359-366 ◽  
Author(s):  
AMBAR N. SENGUPTA
Keyword(s):  
Clifford Algebras ◽  
Quantum Gate ◽  
Finite Geometries ◽  
Fano Plane ◽  
Finite Projective Geometries ◽  
Composite Systems ◽  
Projective Geometries ◽  
Higher Dimensional

Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems.

General Finite Geometries

1971 ◽  
Vol 64 (6) ◽  
pp. 541-545
Author(s):  
Steven H. Heath
Keyword(s):  
High School ◽  
Large Class ◽  
Finite Geometries ◽  
Finite Systems ◽  
Finite Projective Geometries ◽  
Projective Geometries ◽  
The Subject ◽  
And Algebra

In recent years the subject of miniature or finite geometries has aroused some interest. Most studies to date have been concerned with the study of finite projective and affine (Euclidean) geometries, Many teachers have found this material simple enough to present to students studying geometry and algebra in high school. This paper proposes to describe axiomatically a large class of finite systems called finite Bolyai-Lobachevsky geometries. The approach will be geometric instead of algebraic, although to pursue the subject in depth algebraic techniques are needed. The Bolyai-Lobachevsky geometries can be presented in somewhat the same manner as the studies of finite projective geometries.

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