MinT - Best Known (51, 51+29, s)-Nets in Base 81

Best Known (51, 51+29, s)-Nets in Base 81

(51, 51+29, 830)-Net over F₈₁ — Constructive and digital

Digital (51, 80, 830)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]
2. digital (36, 65, 730)-net over F₈₁, using
  - net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
      - the Hermitian function field over F₈₁ [i]

(51, 51+29, 40076)-Net over F₈₁ — Digital

Digital (51, 80, 40076)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+29, large)-Net in Base 81 — Upper bound on s

There is no (51, 80, large)-net in base 81, because

27 times m-reduction [i] would yield (51, 53, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]