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

  • * Corresponding author: Alexander A. Davydov

    * Corresponding author: Alexander A. Davydov 
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  • The length function $ \ell_q(r,R) $ is the smallest length of a $ q $-ary linear code with codimension (redundancy) $ r $ and covering radius $ R $. In this work, new upper bounds on $ \ell_q(tR+1,R) $ are obtained in the following forms:

    $ \begin{equation*} \begin{split} &(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $

    $ \begin{equation*} \begin{split} &(b)\; \ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $

    In the literature, for $ q = (q')^R $ with $ q' $ a prime power, smaller upper bounds are known; however, when $ q $ is an arbitrary prime power, the bounds of this paper are better than the known ones.

    For $ t = 1 $, we use a one-to-one correspondence between $ [n,n-(R+1)]_qR $ codes and $ (R-1) $-saturating $ n $-sets in the projective space $ \mathrm{PG}(R,q) $. A new construction of such saturating sets providing sets of small size is proposed. Then the $ [n,n-(R+1)]_qR $ codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "$ q^m $-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension $ r = tR+1 $, $ t\ge1 $.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 51E21, 51E22.

    Citation:

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  • Figure 1.  The sizes $ \overline{t}(3,q) $ of the smallest known complete arcs in $ \mathrm{PG}(3,q) $ (bottom curve) vs upper bounds of Theorem 6.10 and Theorem 3.3 (top curve) for $ R = 3 $, $ t = 1 $; the sizes and bounds are divided by $ \sqrt[3]{q\ln q} $; $ \lambda = 3 $

    Table 1.  Examples of values connected with upper bounds of Theorem 6.10 and Theorem 3.3 for $ t = 1 $; $ Q_0\in\{5\cdot10^4,15\cdot10^4\} $, $ E = e^{R-1} $

    $ R \\E$ $ \lambda $ $ \Upsilon_{\lambda,R}(E) $ $ Q_{\lambda,R} $ $ C_{\lambda,R} $ $ \Omega_{\lambda,R}(Q_{0})\\Q_0= 5\cdot10^4 $ $ \Omega_{\lambda,R}(Q_0) \\Q_0=15\cdot10^4$ $ D_{\lambda,R} $
    3 2.35 2.25 1007 9.50 6.43 6.17 5.61
    7.39 3 3.67 7186 7.14 5.90 5.60 5
    $ \lambda_{\min}= $3.302 4.44 14974 6.69 5.93 5.58 $ 4.953=D^{\min}_R =1.651R$
    4 2.2 1.91 6826 25.9 18.49 16.42 11.22
    20.1 2.5 2.80 61724 16.5 14.30 8.64
    $ \lambda_{\min}= 4.120$ 12.55 118409572 6.89 $ 5.493=D^{\min}_R =1.373R$
    5 2.3 1.59 21242 84.3 68.53 55.4 23.74
    54.6 2.5 2.22 283935 45.1 17.86
    $ \lambda_{\min}= 4.743$ 28.72 $ 5.929=D^{\min}_R =1.186R$
    6 2.5 1.35 37774 337 304.6 217.7 46.73
    148 $ \lambda_{\min}= 5.277$ 56.67 $ 6.333=D^{\min}_R =1.056R$
    7 2.95 1.80 9125037 265 56.48
    403 $ \lambda_{\min}= 5.765$ 100.5 $ 6.726=D^{\min}_R =0.961R$
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