MinT - Best Known (71−19, 71, s)-Nets in Base 81

Best Known (71−19, 71, s)-Nets in Base 81

(71−19, 71, 59295)-Net over F₈₁ — Constructive and digital

Digital (52, 71, 59295)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 246)-net over F₈₁, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 4, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
    3. digital (0, 9, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
2. digital (36, 55, 59049)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵⁵, 59049, F₈₁, 19, 19) (dual of [(59049, 19), 1121876, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(81⁵⁵, 531442, F₈₁, 19) (dual of [531442, 531387, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(71−19, 71, 3183863)-Net over F₈₁ — Digital

Digital (52, 71, 3183863)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−19, 71, large)-Net in Base 81 — Upper bound on s

There is no (52, 71, large)-net in base 81, because

17 times m-reduction [i] would yield (52, 54, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]