MinT - Best Known (89−15, 89, s)-Nets in Base 9

Best Known (89−15, 89, s)-Nets in Base 9

(89−15, 89, 151842)-Net over F₉ — Constructive and digital

Digital (74, 89, 151842)-net over F₉, using

9¹ times duplication [i] based on digital (73, 88, 151842)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁸⁸, 151842, F₉, 15, 15) (dual of [(151842, 15), 2277542, 16]-NRT-code), using
  - OOA 7-folding and stacking with additional row [i] based on linear OA(9⁸⁸, 1062895, F₉, 15) (dual of [1062895, 1062807, 16]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁸⁸, 1062898, F₉, 15) (dual of [1062898, 1062810, 16]-code), using
      - trace code [i] based on linear OA(81⁴⁴, 531449, F₈₁, 15) (dual of [531449, 531405, 16]-code), using
        constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(81⁴³, 531442, F₈₁, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(81³⁷, 531442, F₈₁, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(81¹, 7, F₈₁, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(89−15, 89, 1062900)-Net over F₉ — Digital

Digital (74, 89, 1062900)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, 1062900, F₉, 15) (dual of [1062900, 1062811, 16]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9⁸⁸, 1062898, F₉, 15) (dual of [1062898, 1062810, 16]-code), using
    - trace code [i] based on linear OA(81⁴⁴, 531449, F₈₁, 15) (dual of [531449, 531405, 16]-code), using
      - constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(81⁴³, 531442, F₈₁, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(81³⁷, 531442, F₈₁, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(81¹, 7, F₈₁, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. linear OA(9⁸⁸, 1062899, F₉, 14) (dual of [1062899, 1062811, 15]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9⁸⁸Â > V_b^sâˆ’1(kâˆ’1)Â = 195 096274 449505 849999 778866 197233 512863 310189 042254 841461 945377 633250 865364 810193 [i]
  3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(89−15, 89, large)-Net in Base 9 — Upper bound on s

There is no (74, 89, large)-net in base 9, because

13 times m-reduction [i] would yield (74, 76, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (89−15, 89, s)-Nets in Base 9

(89−15, 89, 151842)-Net over F9 — Constructive and digital

(89−15, 89, 1062900)-Net over F9 — Digital

(89−15, 89, large)-Net in Base 9 — Upper bound on s

(89−15, 89, 151842)-Net over F₉ — Constructive and digital

(89−15, 89, 1062900)-Net over F₉ — Digital