<?xml version="1.0" encoding="utf-8"?> <feed xmlns="http://www.w3.org/2005/Atom"> <title type="text">Recent zbMATH articles in MSC 94B75</title> <id>https://zbmath.org/atom/cc/94B75</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/" /> <link href="https://zbmath.org/atom/cc/94B75" rel="self" /> <generator>Werkzeug</generator> <entry xml:base="https://zbmath.org/atom/cc/94B75"> <title type="text">New bounds for covering codes of radius 3 and codimension \(3 t + 1\)</title> <id>https://zbmath.org/1553.94114</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.94114" /> <author> <name>&quot;Davydov, Alexander A.&quot;</name> <uri>https://zbmath.org/authors/?q=ai:davydov.alexander-a</uri> </author> <author> <name>&quot;Marcugini, Stefano&quot;</name> <uri>https://zbmath.org/authors/?q=ai:marcugini.stefano</uri> </author> <author> <name>&quot;Pambianco, Fernanda&quot;</name> <uri>https://zbmath.org/authors/?q=ai:pambianco.fernanda</uri> </author> <content type="text">Summary: The smallest possible length of a \(q\)-ary linear code of covering radius \(R\) and codimension (redundancy) \(r\) is called the length function and is denoted by \(\ell_q(r, R)\). In this work, for \(q\) an arbitrary prime power, we obtain the following new constructive upper bounds on \(\ell_q(3 t + 1, 3)\): \begin{itemize} \item \(\ell_q ( r , 3 ) \lessapprox \sqrt[3]{k} \cdot q^{( r - 3 ) / 3} \cdot \sqrt[3]{ \ln q} \), \( r = 3 t + 1 \), \(t \geq 1 \), \( q \geq \lceil \mathcal{W} ( k ) \rceil \), \( 18 &lt; k \leq 20.339 \), \( \mathcal{W} ( k ) \text{ is a decreasing function of } k \); \item \(\ell_q ( r , 3 ) \lessapprox \sqrt[3]{18} \cdot q^{( r - 3 ) / 3} \cdot \sqrt[3]{ \ln q} \), \(r = 3 t + 1 \), \(t \geq 1 \), \(q \text{ large enough} \). \end{itemize} For \(t = 1\), we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space \(\mathrm{PG}(3, q)\). A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called ``\(q^m\)-concatenating constructions'') to obtain infinite families of codes with radius 3 and growing codimension \(r = 3 t + 1\), \(t \geq 1\). The new bounds are essentially better than the known ones.</content> </entry> </feed>