MinT - Best Known (57−33, 57, s)-Nets in Base 128

Best Known (57−33, 57, s)-Nets in Base 128

(57−33, 57, 408)-Net over F₁₂₈ — Constructive and digital

Digital (24, 57, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 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 (5, 38, 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]

(57−33, 57, 514)-Net in Base 128 — Constructive

(24, 57, 514)-net in base 128, using

128¹ times duplication [i] based on (23, 56, 514)-net in base 128, using
- base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 16, 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. digital (0, 33, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(57−33, 57, 587)-Net over F₁₂₈ — Digital

Digital (24, 57, 587)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−33, 57, 1270478)-Net in Base 128 — Upper bound on s

There is no (24, 57, 1270479)-net in base 128, because

1 times m-reduction [i] would yield (24, 56, 1270479)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 10086 988021 990797 800614 701347 109680 689025 669057 144074 550339 786594 999187 277137 097479 333568 488624 086275 085747 840969 334028 > 128⁵⁶ [i]