On cutting blocking sets and their codes

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.

Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

<p style='text-indent:20px;'>The length function <inline-formula><tex-math id="M3">\begin{document}$ \ell_q(r,R) $\end{document}</tex-math></inline-formula> is the smallest length of a <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary linear code with codimension (redundancy) <inline-formula><tex-math id="M5">\begin{document}$ r $\end{document}</tex-math></inline-formula> and covering radius <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In this work, new upper bounds on <inline-formula><tex-math id="M7">\begin{document}$ \ell_q(tR+1,R) $\end{document}</tex-math></inline-formula> are obtained in the following forms:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} &amp;(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} &amp;(b)\; \ell_q(r,R)&lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>In the literature, for <inline-formula><tex-math id="M8">\begin{document}$ q = (q')^R $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ q' $\end{document}</tex-math></inline-formula> a prime power, smaller upper bounds are known; however, when <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> is an arbitrary prime power, the bounds of this paper are better than the known ones.</p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M11">\begin{document}$ t = 1 $\end{document}</tex-math></inline-formula>, we use a one-to-one correspondence between <inline-formula><tex-math id="M12">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes and <inline-formula><tex-math id="M13">\begin{document}$ (R-1) $\end{document}</tex-math></inline-formula>-saturating <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>-sets in the projective space <inline-formula><tex-math id="M15">\begin{document}$ \mathrm{PG}(R,q) $\end{document}</tex-math></inline-formula>. A new construction of such saturating sets providing sets of small size is proposed. Then the <inline-formula><tex-math id="M16">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "<inline-formula><tex-math id="M17">\begin{document}$ q^m $\end{document}</tex-math></inline-formula>-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension <inline-formula><tex-math id="M18">\begin{document}$ r = tR+1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}$ t\ge1 $\end{document}</tex-math></inline-formula>.</p>

The geometry of Hermitian self-orthogonal codes

We prove that if $$n >k^2$$ n > k 2 then a k-dimensional linear code of length n over $${\mathbb F}_{q^2}$$ F q 2 has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.

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

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.

On some punctured codes of ℤq-Simplex codes

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

METHOD OF PROTECTING INFORMATION PROCESSING DEVICES OF DIGITAL DATA TRANSMISSION SYSTEMS FROM SINGLE BYTE ERRORS

Предложена регулярная процедура адаптации кода, исправляющего одиночные байтовые ошибки, с целью обнаружения и коррекции байтовых ошибок в арифметико-логических устройствах(АЛУ) процессоров информационно-измерительных систем. Выявлены закономерности, определяющие соотношения между арифметико-логическими операциями и значениями контрольных разрядов линейного кода относительно данных операций. Показано, что эти закономерности позволяют сформулировать правила получения значений поправок к контрольным разрядам кода для обнаружения и коррекции одиночных байтовых ошибок при использовании алгебраического линейного кода с синдромным декодированием. Предлагаемый метод защиты от одиночных байтовых ошибок устройств обработки информации позволяет минимизировать влияние кодирования (декодирования) информации на быстродействие АЛУ процессора за счет замены циклической процедуры кодирования информации на алгебраическую с синдромным декодированием. A regular procedure for adapting the code that corrects single bit errors for detecting and correcting byte errors in arithmetic-logic units (ALU) of information and measurement system processors is proposed. The regularities that determine the relations between arithmetic logical operations and the values of the control digits of the linear code relative to these operations are revealed. It is shown that these regularities allow us to formulate rules for obtaining correction values of code control bits for detecting and correcting single byte errors when using an algebraic linear code with syndrome decoding. The proposed method of protection against single bit errors of information processing devices allows minimizing the impact of encoding (decoding) information on the performance of the ALU processor by replacing the cyclic information encoding procedure with an algebraic one with syndrome decoding.

Projective MDS Codes Over GF(27)‎

MDS code is a linear code that achieves equality in the Singleton bound, and projective MDS (PG-MDS) is MDS code with independents property of any two columns of its generator matrix. In this paper, elementary methods for modifying a PG-MDS code of dimensions 2, 3, as extending and lengthening, in order to find new incomplete PG-MDS codes have been used over . Also, two complete PG-MDS codes over of length and 28 have been found.