MinT - Best Known (27, 27+40, s)-Nets in Base 128

Best Known (27, 27+40, s)-Nets in Base 128

(27, 27+40, 384)-Net over F₁₂₈ — Constructive and digital

Digital (27, 67, 384)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (27, 69, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 24, 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. digital (3, 45, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(27, 27+40, 407)-Net in Base 128 — Constructive

(27, 67, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 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. (6, 46, 257)-net in base 128, using
  - 2 times m-reduction [i] based on (6, 48, 257)-net in base 128, using
    - base change [i] based on digital (0, 42, 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]

(27, 27+40, 525)-Net over F₁₂₈ — Digital

Digital (27, 67, 525)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+40, 749290)-Net in Base 128 — Upper bound on s

There is no (27, 67, 749291)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 305706 310495 036817 176262 872118 918379 536328 714746 684838 925020 101345 757877 258703 333774 922238 624390 459500 779350 091852 883944 481134 291549 078308 > 128⁶⁷ [i]