MinT - Best Known (19, 19+37, s)-Nets in Base 128

Best Known (19, 19+37, s)-Nets in Base 128

(19, 19+37, 288)-Net over F₁₂₈ — Constructive and digital

Digital (19, 56, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 56, 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]

(19, 19+37, 386)-Net over F₁₂₈ — Digital

Digital (19, 56, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 56, 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]

(19, 19+37, 513)-Net in Base 128

(19, 56, 513)-net in base 128, using

t-expansion [i] based on (17, 56, 513)-net in base 128, using
- 16 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]

(19, 19+37, 163305)-Net in Base 128 — Upper bound on s

There is no (19, 56, 163306)-net in base 128, because

1 times m-reduction [i] would yield (19, 55, 163306)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 809424 812604 151502 535104 600924 878578 476583 780488 789742 053739 018240 226069 440538 505908 268170 287134 800380 964335 465512 > 128⁵⁵ [i]