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New bounds for covering codes of radius 3 and codimension $ 3t+1 $

  • * Corresponding author: Fernanda Pambianco

    * Corresponding author: Fernanda Pambianco
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  • 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(3t+1,3) $:

    $ \begin{equation*} \begin{split} &\bullet\; \ell_q(r,3)\lessapprox \sqrt[3]{k}\cdot q^{(r-3)/3}\cdot\sqrt[3]{\ln q}, \; r = 3t+1, \; t\ge1, \; q\ge\lceil \mathcal{W}(k)\rceil, \\ &\phantom{\bullet\; }18 <k\le20.339, \; \mathcal{W}(k){\text{ is a decreasing function of }}k ;\\ &\bullet\; \ell_q(r,3)\lessapprox \sqrt[3]{18}\cdot q^{(r-3)/3}\cdot\sqrt[3]{\ln q}, \; r = 3t+1, \; t\ge1, \; q{\text{ large enough}}. \end{split} \end{equation*} $

    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 = 3t + 1 $, $ t\ge1 $. The new bounds are essentially better than the known ones.

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

    Citation:

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  • Figure 1.  Upper bounds on the length function $ \ell_q(4,3) $ divided by $ \sqrt[3]{q\ln q} $: implicit Bound A (3.5)–(3.6) for $ 8233\le q<10^5 $ (the second curve), computer Bound E (3.25) for $ 13\le q\le8231 $ (the bottom curve) vs the known bound (5.1)–(5.2) for $ 14983\le q<10^5 $ (the top curve)

    Figure 2.  Upper bounds on the length function $ \ell_q(4,3) $ divided by $ \sqrt[3]{q\ln q} $: implicit Bound A (3.5)–(3.6) (the bottom, solid curve), implicit Bound B (3.10)–(3.11) (the second, dashed curve), and explicit Bound C (points $ 1 $, $ 2 $, and $ 3 $ correspond to $ n^{\text{C}}_{4, q}(k)/\sqrt[3]{ q\ln q} $ with $ k = 20.339, \;20 $, and $ 19.7 $, by Table 1) vs the known bound (5.1)–(5.2) (the top, solid curve), $ 10^5< q<5\cdot10^6 $

    Figure 3.  The ratio $ n^{knw}_{4, q}/n^{\mathrm{A}}_{4, q} $ of the known [17] upper bound $ n^{knw}_{4, q} $ (5.1)–(5.2) on the length function $ \ell_q(4,3) $ and the new one $ n^{\mathrm{A}}_{4, q} $ (3.5)–(3.6), $ 14983\le q<5\cdot10^6 $

    Table 1.  Values of $ \lceil \mathcal{W}(k)\rceil $ for $ 18.0001\le k\le 20.340 $ and values of $ n^{\text{C}}_{4, q}(k)/\sqrt[3]{ q\ln q} $, $ n^{knw}_{4, q}/\sqrt[3]{q\ln q} $ (the known bound), $ n^{knw}_{4, q}/n^{\text{C}}_{4, q}(k) $ for $ q = \lceil \mathcal{W}(k)\rceil $, $ 18.0001\le k\le 20.339 $. ($ \mathcal{V} = 1516750 $)

    $ k $ $ \lceil \mathcal{W}(k)\rceil $ $ \frac{n^{\text{C}}_{4, q}(k)}{\sqrt[3]{ q\ln q}} $ $ \frac{n^{knw}_{4, q}}{\sqrt[3]{q\ln q}} $ $ \frac{n^{knw}_{4, q}}{n^{\text{C}}_{4, q}(k)} $
    20.340 $ 1515738\thickapprox1.516\cdot10^{6}< \mathcal{V} $
    20.339 $ 1517567\thickapprox1.518\cdot10^{6}> \mathcal{V} $ 2.7368 5.2500 1.9183
    20.335 $ 1524915\thickapprox1.525\cdot10^{6}> \mathcal{V} $ 2.7367 5.2495 1.9182
    20 $ 2374364\thickapprox2.374\cdot10^{6}> \mathcal{V} $ 2.7205 5.2087 1.9146
    19.7 $ 3820987\thickapprox3.821\cdot10^{6}> \mathcal{V} $ 2.7059 5.1716 1.9112
    19 $ 19178705\thickapprox1.918\cdot10^{7}> \mathcal{V} $ 2.6713 5.0828 1.9027
    18.5 $ 171670620\thickapprox1.717\cdot10^{8}> \mathcal{V} $ 2.6461 5.0180 1.8963
    18.1 $ 30640000001\thickapprox3.064\cdot10^{10}> \mathcal{V} $ 2.6258 4.9659 1.8912
    18.05 $ 294427001643\thickapprox2.944\cdot10^{11}> \mathcal{V} $ 2.6233 4.9593 1.8905
    18.01 $ 52060446118120\thickapprox5.206\cdot10^{13}> \mathcal{V} $ 2.6212 4.9542 1.8900
    18.001 $ \thickapprox7.880\cdot 10^{16}> \mathcal{V} $ 2.6208 4.9530 1.8899
    18.0001 $ \thickapprox1.109\cdot 10^{20}> \mathcal{V} $ 2.6207 4.9529 1.8899
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