MinT - Best Known (25, 62, s)-Nets in Base 128

Best Known (25, 62, s)-Nets in Base 128

(25, 62, 384)-Net over F₁₂₈ — Constructive and digital

Digital (25, 62, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (25, 63, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 22, 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, 41, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(25, 62, 407)-Net in Base 128 — Constructive

(25, 62, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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, 43, 257)-net in base 128, using
  - 5 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]

(25, 62, 513)-Net over F₁₂₈ — Digital

Digital (25, 62, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 62, 513)-net over F₁₂₈, using
- net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]

(25, 62, 823042)-Net in Base 128 — Upper bound on s

There is no (25, 62, 823043)-net in base 128, because

1 times m-reduction [i] would yield (25, 61, 823043)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 346 587979 613504 281962 397291 674940 622771 560331 509472 897681 034726 792549 711297 699210 949013 664208 475872 412485 573678 176286 087225 951120 > 128⁶¹ [i]