MinT - Best Known (58−37, 58, s)-Nets in Base 128

Best Known (58−37, 58, s)-Nets in Base 128

(58−37, 58, 321)-Net over F₁₂₈ — Constructive and digital

Digital (21, 58, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 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. digital (3, 40, 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]

(58−37, 58, 386)-Net over F₁₂₈ — Digital

Digital (21, 58, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 58, 386)-net over F₁₂₈, using
- net from sequence [i] based on digital (15, 385)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(58−37, 58, 513)-Net in Base 128

(21, 58, 513)-net in base 128, using

t-expansion [i] based on (17, 58, 513)-net in base 128, using
- 14 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(58−37, 58, 279991)-Net in Base 128 — Upper bound on s

There is no (21, 58, 279992)-net in base 128, because

1 times m-reduction [i] would yield (21, 57, 279992)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 291184 892957 724203 175295 412111 472406 186860 852498 049610 709746 252311 291356 074932 582899 747023 238032 752412 240780 120167 981335 > 128⁵⁷ [i]