MinT - Best Known (80−10, 80, s)-Nets in Base 9

Best Known (80−10, 80, s)-Nets in Base 9

(80−10, 80, 3355456)-Net over F₉ — Constructive and digital

Digital (70, 80, 3355456)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (64, 74, 3355440)-net over F₉, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81³⁷, 1677720, F₈₁, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(81³⁷, 8388600, F₈₁, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(81³⁷, large, F₈₁, 10) (dual of [large, large−37, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(80−10, 80, large)-Net over F₉ — Digital

Digital (70, 80, large)-net over F₉, using

t-expansion [i] based on digital (69, 80, large)-net over F₉, using
- 1 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]

(80−10, 80, large)-Net in Base 9 — Upper bound on s

There is no (70, 80, large)-net in base 9, because

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