MinT - Best Known (21, 21+43, s)-Nets in Base 128

Best Known (21, 21+43, s)-Nets in Base 128

(21, 21+43, 288)-Net over F₁₂₈ — Constructive and digital

Digital (21, 64, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 64, 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+43, 386)-Net over F₁₂₈ — Digital

Digital (21, 64, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 64, 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+43, 513)-Net in Base 128

(21, 64, 513)-net in base 128, using

t-expansion [i] based on (17, 64, 513)-net in base 128, using
- 8 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]

(21, 21+43, 143313)-Net in Base 128 — Upper bound on s

There is no (21, 64, 143314)-net in base 128, because

1 times m-reduction [i] would yield (21, 63, 143314)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 5 678610 326890 249205 270052 011903 661362 458550 261190 974605 450123 130276 785179 435074 932171 965734 096255 139988 647175 532576 468794 285615 378096 > 128⁶³ [i]