MinT - Best Known (69−51, 69, s)-Nets in Base 128

Best Known (69−51, 69, s)-Nets in Base 128

(69−51, 69, 288)-Net over F₁₂₈ — Constructive and digital

Digital (18, 69, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 69, 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]

(69−51, 69, 386)-Net over F₁₂₈ — Digital

Digital (18, 69, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 69, 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]

(69−51, 69, 513)-Net in Base 128

(18, 69, 513)-net in base 128, using

t-expansion [i] based on (17, 69, 513)-net in base 128, using
- 3 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]

(69−51, 69, 43182)-Net in Base 128 — Upper bound on s

There is no (18, 69, 43183)-net in base 128, because

1 times m-reduction [i] would yield (18, 68, 43183)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 195134 301445 232759 325799 153651 376866 761370 084237 417773 519108 840985 232534 884308 675842 863910 989923 605493 246133 314565 295732 725613 317191 594306 367612 > 128⁶⁸ [i]