scholarly journals An Extension of the Brouwer–Zimmermann Algorithm for Calculating the Minimum Weight of a Linear Code

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2354
Author(s):  
Stefka Bouyuklieva ◽  
Iliya Bouyukliev
Keyword(s):  
Finite Field ◽  
Linear Code ◽  
Software Package ◽  
Minimum Weight

A modification of the Brouwer–Zimmermann algorithm for calculating the minimum weight of a linear code over a finite field is presented. The aim was to reduce the number of codewords for consideration. The reduction is significant in cases where the length of a code is not divisible by its dimensions. The proposed algorithm can also be used to find all codewords of weight less than a given constant. The algorithm is implemented in the software package QextNewEdition.

Estimates for distribution of the minimal distance of a random linear code

2016 ◽  
Vol 26 (4) ◽  
Author(s):  
Viktor A. Kopyttcev ◽  
Vladimir G. Mikhailov
Keyword(s):  
Distribution Function ◽  
Finite Field ◽  
Linear Code ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Minimal Distance ◽  
Asymptotic Estimates

AbstractThe distribution function of the minimum distance (the minimum weight of nonzero codewords) of a random linear code over a finite field is studied. Expicit bounds in the form of inequalities and asymptotic estimates for this distribution function are obtained.

Isometry groups of additive codes over finite fields

2018 ◽  
Vol 17 (10) ◽  
pp. 1850198 ◽  
Author(s):  
Jay A. Wood
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Linear Code ◽  
Necessary Conditions ◽  
Isometry Groups ◽  
Additive Codes ◽  
Prime Fields

When [Formula: see text] is a linear code over a finite field [Formula: see text], every linear Hamming isometry of [Formula: see text] to itself is the restriction of a linear Hamming isometry of [Formula: see text] to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping [Formula: see text] to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.The monomial transformations mapping [Formula: see text] to itself form a group [Formula: see text], and the additive Hamming isometries form a larger group [Formula: see text]: [Formula: see text]. The main result says that these two subgroups can be as different as possible: for any two subgroups [Formula: see text], subject to some natural necessary conditions, there exists an additive code [Formula: see text] such that [Formula: see text] and [Formula: see text].

scholarly journals On cutting blocking sets and their codes

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniele Bartoli ◽  
Antonio Cossidente ◽  
Giuseppe Marino ◽  
Francesco Pavese
Keyword(s):  
Finite Field ◽  
Pairwise Disjoint ◽  
Linear Code ◽  
Linear Codes ◽  
Blocking Set ◽  
Minimum Length ◽  
Blocking Sets ◽  
Minimal Linear Code ◽  
Dimensional Projective Space ◽  
Line Spread

Abstract Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

scholarly journals Nonexistence of some ternary linear codes with minimum weight -2 modulo 9

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Toshiharu Sawashima ◽  
Tatsuya Maruta
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Minimum Weight ◽  
Extension Theorem ◽  
Minimum Length ◽  
Geometric Methods ◽  
Optimal Linear ◽  
Ternary Linear Codes ◽  
Optimal Linear Codes

<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id="M3">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id="M9">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id="M10">\begin{document}$ n_3(6,d) $\end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id="M11">\begin{document}$ d $\end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>

Quanitative aspects of electron diffraction using electron holography

1995 ◽  
Vol 53 ◽  
pp. 616-617
Author(s):  
E. Völkl ◽  
L.F. Allard ◽  
B. Frost ◽  
T.A. Nolan
Keyword(s):  
Electron Beam ◽  
Electron Diffraction ◽  
Software Package ◽  
Spatial Coherence ◽  
Electron Holography ◽  
Image Intensity ◽  
Interference Fringes ◽  
Complex Image ◽  
The Difference ◽  
Recorded Image

Off-axis electron holography has the well known ability to preserve the complex image wave within the final, recorded image. This final image described by I(x,y) = I(r) contains contributions from the image intensity of the elastically scattered electrons IeI (r) = |A(r) exp (iΦ(r)) |, the contributions from the inelastically scattered electrons IineI (r), and the complex image wave Ψ = A(r) exp(iΦ(r)) as:(1) I(r) = IeI (r) + Iinel (r) + μ A(r) cos(2π Δk r + Φ(r))where the constant μ describes the contrast of the interference fringes which are related to the spatial coherence of the electron beam, and Φk is the resulting vector of the difference of the wavefront vectors of the two overlaping beams. Using a software package like HoloWorks, the complex image wave Ψ can be extracted.

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