MinT - Best Known (52, 52+28, s)-Nets in Base 81

Best Known (52, 52+28, s)-Nets in Base 81

(52, 52+28, 846)-Net over F₈₁ — Constructive and digital

Digital (52, 80, 846)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]
2. digital (36, 64, 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]

(52, 52+28, 61685)-Net over F₈₁ — Digital

Digital (52, 80, 61685)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+28, large)-Net in Base 81 — Upper bound on s

There is no (52, 80, large)-net in base 81, because

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