MinT - Best Known (75−43, 75, s)-Nets in Base 128

Best Known (75−43, 75, s)-Nets in Base 128

(75−43, 75, 438)-Net over F₁₂₈ — Constructive and digital

Digital (32, 75, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (32, 76, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 23, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (9, 53, 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]

(75−43, 75, 515)-Net in Base 128 — Constructive

(32, 75, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 24, 257)-net in base 128, using
  - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. (8, 51, 258)-net in base 128, using
  - 5 times m-reduction [i] based on (8, 56, 258)-net in base 128, using
    - base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(75−43, 75, 774)-Net over F₁₂₈ — Digital

Digital (32, 75, 774)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−43, 75, 1820090)-Net in Base 128 — Upper bound on s

There is no (32, 75, 1820091)-net in base 128, because

1 times m-reduction [i] would yield (32, 74, 1820091)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858104 759332 860621 689845 404792 412686 348851 836057 158467 094451 746426 564225 728488 391845 411484 402977 787088 837118 969611 824573 724605 429220 899349 263091 148676 920004 > 128⁷⁴ [i]