MinT - Best Known (75−40, 75, s)-Nets in Base 128

Best Known (75−40, 75, s)-Nets in Base 128

(75−40, 75, 504)-Net over F₁₂₈ — Constructive and digital

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]

(75−40, 75, 547)-Net in Base 128 — Constructive

(35, 75, 547)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (6, 26, 259)-net in base 128, using
  - 6 times m-reduction [i] based on (6, 32, 259)-net in base 128, using
    - base change [i] based on digital (2, 28, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 49, 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]

(75−40, 75, 1387)-Net over F₁₂₈ — Digital

Digital (35, 75, 1387)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−40, 75, 5218420)-Net in Base 128 — Upper bound on s

There is no (35, 75, 5218421)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 109 836868 114616 914772 207905 008885 558004 832747 310898 006487 284744 995941 661548 213878 413029 076216 429543 918140 942308 718644 425491 092077 411589 358735 012037 742386 989171 > 128⁷⁵ [i]