MinT - Best Known (70−10, 70, s)-Nets in Base 9

Best Known (70−10, 70, s)-Nets in Base 9

(70−10, 70, 1677730)-Net over F₉ — Constructive and digital

Digital (60, 70, 1677730)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (55, 65, 1677720)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁶⁵, 1677720, F₉, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(9⁶⁵, 8388600, F₉, 10) (dual of [8388600, 8388535, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁶⁵, large, F₉, 10) (dual of [large, large−65, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(70−10, 70, large)-Net over F₉ — Digital

Digital (60, 70, large)-net over F₉, using

9⁵ times duplication [i] based on digital (55, 65, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶⁵, large, F₉, 10) (dual of [large, large−65, 11]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(70−10, 70, large)-Net in Base 9 — Upper bound on s

There is no (60, 70, large)-net in base 9, because

8 times m-reduction [i] would yield (60, 62, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]