MinT - Best Known (21, 21+53, s)-Nets in Base 128

Best Known (21, 21+53, s)-Nets in Base 128

(21, 21+53, 288)-Net over F₁₂₈ — Constructive and digital

Digital (21, 74, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 74, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(21, 21+53, 386)-Net over F₁₂₈ — Digital

Digital (21, 74, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 74, 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]

(21, 21+53, 513)-Net in Base 128

(21, 74, 513)-net in base 128, using

128² times duplication [i] based on (19, 72, 513)-net in base 128, using
- t-expansion [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]

(21, 21+53, 68518)-Net in Base 128 — Upper bound on s

There is no (21, 74, 68519)-net in base 128, because

1 times m-reduction [i] would yield (21, 73, 68519)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6705 907439 546536 506711 077435 709488 888699 706094 168311 125752 503765 194553 755176 376281 934324 314751 997599 907182 003613 976201 342268 792242 601486 011257 286256 017532 > 128⁷³ [i]