MinT - Best Known (22, 22+53, s)-Nets in Base 128

Best Known (22, 22+53, s)-Nets in Base 128

(22, 22+53, 288)-Net over F₁₂₈ — Constructive and digital

Digital (22, 75, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 75, 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]

(22, 22+53, 386)-Net over F₁₂₈ — Digital

Digital (22, 75, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 75, 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]

(22, 22+53, 513)-Net in Base 128

(22, 75, 513)-net in base 128, using

128³ 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]

(22, 22+53, 82578)-Net in Base 128 — Upper bound on s

There is no (22, 75, 82579)-net in base 128, because

1 times m-reduction [i] would yield (22, 74, 82579)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858271 818608 196008 460016 464258 212554 378825 405925 557346 524909 457752 409540 645993 992849 762450 617706 880687 127228 838262 929105 901270 911442 871521 120783 223885 826576 > 128⁷⁴ [i]