MinT - Best Known (70, 78, s)-Nets in Base 9

Best Known (70, 78, s)-Nets in Base 9

(70, 78, 4725744)-Net over F₉ — Constructive and digital

Digital (70, 78, 4725744)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (16, 20, 531444)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9²⁰, 531444, F₉, 4, 4) (dual of [(531444, 4), 2125756, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(9²⁰, 531444, F₉, 3, 4) (dual of [(531444, 3), 1594312, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(9²⁰, 1062888, F₉, 4) (dual of [1062888, 1062868, 5]-code), using
        trace code [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (50, 58, 4194300)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁵⁸, 4194300, F₉, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(9⁵⁸, 8388601, F₉, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁵⁸, 8388602, F₉, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(81²⁹, 4194301, F₈₁, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81²⁹, 8388602, F₈₁, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(81²⁹, large, F₈₁, 8) (dual of [large, large−29, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(70, 78, large)-Net over F₉ — Digital

Digital (70, 78, large)-net over F₉, using

t-expansion [i] based on digital (69, 78, large)-net over F₉, using
- 3 times m-reduction [i] based on digital (69, 81, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸¹, large, F₉, 12) (dual of [large, large−81, 13]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(70, 78, large)-Net in Base 9 — Upper bound on s

There is no (70, 78, large)-net in base 9, because

6 times m-reduction [i] would yield (70, 72, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (70, 78, s)-Nets in Base 9

(70, 78, 4725744)-Net over F9 — Constructive and digital

(70, 78, large)-Net over F9 — Digital

(70, 78, large)-Net in Base 9 — Upper bound on s

(70, 78, 4725744)-Net over F₉ — Constructive and digital

(70, 78, large)-Net over F₉ — Digital