MinT - Best Known (21, 21+45, s)-Nets in Base 128

Best Known (21, 21+45, s)-Nets in Base 128

(21, 21+45, 288)-Net over F₁₂₈ — Constructive and digital

Digital (21, 66, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 66, 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+45, 386)-Net over F₁₂₈ — Digital

Digital (21, 66, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 66, 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+45, 513)-Net in Base 128

(21, 66, 513)-net in base 128, using

t-expansion [i] based on (17, 66, 513)-net in base 128, using
- 6 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+45, 119910)-Net in Base 128 — Upper bound on s

There is no (21, 66, 119911)-net in base 128, because

1 times m-reduction [i] would yield (21, 65, 119911)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 93038 821783 288180 799456 064176 175778 334820 197144 869982 812826 778701 218786 981507 222063 257702 071549 163926 571593 880163 001397 132405 764659 396266 > 128⁶⁵ [i]