MinT - Best Known (75−39, 75, s)-Nets in Base 128

Best Known (75−39, 75, s)-Nets in Base 128

(75−39, 75, 504)-Net over F₁₂₈ — Constructive and digital

Digital (36, 75, 504)-net over F₁₂₈, using

t-expansion [i] based on digital (35, 75, 504)-net over F₁₂₈, using
- 2 times m-reduction [i] based on digital (35, 77, 504)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 26, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]
    2. digital (9, 51, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(75−39, 75, 549)-Net in Base 128 — Constructive

(36, 75, 549)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (8, 27, 261)-net in base 128, using
  - 5 times m-reduction [i] based on (8, 32, 261)-net in base 128, using
    - base change [i] based on digital (4, 28, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 48, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(75−39, 75, 1725)-Net over F₁₂₈ — Digital

Digital (36, 75, 1725)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−39, 75, large)-Net in Base 128 — Upper bound on s

There is no (36, 75, large)-net in base 128, because

37 times m-reduction [i] would yield (36, 38, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]