MinT - Best Known (88−12, 88, s)-Nets in Base 9

Best Known (88−12, 88, s)-Nets in Base 9

(88−12, 88, 1398116)-Net over F₉ — Constructive and digital

Digital (76, 88, 1398116)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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 (69, 81, 1398100)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁸¹, 1398100, F₉, 12, 12) (dual of [(1398100, 12), 16777119, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(9⁸¹, 8388600, F₉, 12) (dual of [8388600, 8388519, 13]-code), using
      - discarding factors / shortening the dual code 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]

(88−12, 88, large)-Net over F₉ — Digital

Digital (76, 88, large)-net over F₉, using

1 times m-reduction [i] based on digital (76, 89, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(88−12, 88, large)-Net in Base 9 — Upper bound on s

There is no (76, 88, large)-net in base 9, because

10 times m-reduction [i] would yield (76, 78, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]