MinT - Best Known (68−35, 68, s)-Nets in Base 128

Best Known (68−35, 68, s)-Nets in Base 128

(68−35, 68, 504)-Net over F₁₂₈ — Constructive and digital

Digital (33, 68, 504)-net over F₁₂₈, using

3 times m-reduction [i] based on digital (33, 71, 504)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 24, 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, 47, 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]

(68−35, 68, 549)-Net in Base 128 — Constructive

(33, 68, 549)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (7, 24, 261)-net in base 128, using
  - base change [i] based on digital (4, 21, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (9, 44, 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]

(68−35, 68, 1763)-Net over F₁₂₈ — Digital

Digital (33, 68, 1763)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−35, 68, large)-Net in Base 128 — Upper bound on s

There is no (33, 68, large)-net in base 128, because

33 times m-reduction [i] would yield (33, 35, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]