MinT - Best Known (30, 76, s)-Nets in Base 128

Best Known (30, 76, s)-Nets in Base 128

(30, 76, 384)-Net over F₁₂₈ — Constructive and digital

Digital (30, 76, 384)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (30, 78, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 27, 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, 51, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(30, 76, 386)-Net in Base 128 — Constructive

(30, 76, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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, 53, 257)-net in base 128, using
  - 3 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]

(30, 76, 609)-Net over F₁₂₈ — Digital

Digital (30, 76, 609)-net over F₁₂₈, using

net from sequence [i] based on digital (30, 608)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 30 and N(F) ≥ 609, using
  - a curve from the manYPoints database [i]

(30, 76, 681710)-Net in Base 128 — Upper bound on s

There is no (30, 76, 681711)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 14059 242556 069375 998328 875729 914490 293132 790026 324607 149947 102974 906929 516328 872712 703505 768678 181315 379853 469951 958052 220258 101728 286865 221595 274880 866608 556848 > 128⁷⁶ [i]