MinT - Best Known (32, 32+45, s)-Nets in Base 128

Best Known (32, 32+45, s)-Nets in Base 128

(32, 32+45, 438)-Net over F₁₂₈ — Constructive and digital

Digital (32, 77, 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, 54, 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]

(32, 32+45, 449)-Net in Base 128 — Constructive

(32, 77, 449)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]
2. (7, 52, 257)-net in base 128, using
  - 4 times m-reduction [i] based on (7, 56, 257)-net in base 128, using
    - base change [i] based on digital (0, 49, 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]

(32, 32+45, 684)-Net over F₁₂₈ — Digital

Digital (32, 77, 684)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 32+45, 1356745)-Net in Base 128 — Upper bound on s

There is no (32, 77, 1356746)-net in base 128, because

1 times m-reduction [i] would yield (32, 76, 1356746)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 14059 132630 498342 864153 110281 666330 936938 618798 298960 117415 230009 922058 316876 268386 892352 985711 147051 046344 298509 028623 273045 630413 485466 950828 286668 933223 426704 > 128⁷⁶ [i]