MinT - Best Known (91−18, 91, s)-Nets in Base 9

Best Known (91−18, 91, s)-Nets in Base 9

(91−18, 91, 6581)-Net over F₉ — Constructive and digital

Digital (73, 91, 6581)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (62, 80, 6561)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁸⁰, 6561, F₉, 18, 18) (dual of [(6561, 18), 118018, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(9⁸⁰, 59049, F₉, 18) (dual of [59049, 58969, 19]-code), using
      - 1 times truncation [i] based on linear OA(9⁸¹, 59050, F₉, 19) (dual of [59050, 58969, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 59050Â | 9¹⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(91−18, 91, 115051)-Net over F₉ — Digital

Digital (73, 91, 115051)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−18, 91, large)-Net in Base 9 — Upper bound on s

There is no (73, 91, large)-net in base 9, because

16 times m-reduction [i] would yield (73, 75, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]