MinT - Best Known (74−45, 74, s)-Nets in Base 128

Best Known (74−45, 74, s)-Nets in Base 128

(74−45, 74, 384)-Net over F₁₂₈ — Constructive and digital

Digital (29, 74, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (29, 75, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 26, 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, 49, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(74−45, 74, 386)-Net in Base 128 — Constructive

(29, 74, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [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]

(74−45, 74, 577)-Net over F₁₂₈ — Digital

Digital (29, 74, 577)-net over F₁₂₈, using

t-expansion [i] based on digital (28, 74, 577)-net over F₁₂₈, using
- net from sequence [i] based on digital (28, 576)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577, using
    - a curve from the manYPoints database [i]

(74−45, 74, 700081)-Net in Base 128 — Upper bound on s

There is no (29, 74, 700082)-net in base 128, because

1 times m-reduction [i] would yield (29, 73, 700082)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6704 045996 005119 787413 740847 425336 486193 221674 875830 687357 991665 881067 982659 376163 523086 067657 753544 482706 869745 212569 864241 081476 087076 714509 168115 933712 > 128⁷³ [i]