MinT - Best Known (82−31, 82, s)-Nets in Base 81

Best Known (82−31, 82, s)-Nets in Base 81

(82−31, 82, 812)-Net over F₈₁ — Constructive and digital

Digital (51, 82, 812)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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 (36, 67, 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]

(82−31, 82, 24802)-Net over F₈₁ — Digital

Digital (51, 82, 24802)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−31, 82, large)-Net in Base 81 — Upper bound on s

There is no (51, 82, large)-net in base 81, because

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