MinT - Best Known (15, 56, s)-Nets in Base 128

Best Known (15, 56, s)-Nets in Base 128

(15, 56, 288)-Net over F₁₂₈ — Constructive and digital

Digital (15, 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]

(15, 56, 386)-Net over F₁₂₈ — Digital

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]

(15, 56, 513)-Net in Base 128

(15, 56, 513)-net in base 128, using

base change [i] based on digital (8, 49, 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]

(15, 56, 40759)-Net in Base 128 — Upper bound on s

There is no (15, 56, 40760)-net in base 128, because

1 times m-reduction [i] would yield (15, 55, 40760)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 829897 840351 967299 637369 561838 269284 918026 626828 640821 605571 600396 383419 736838 768380 198778 128183 629081 037188 198157 > 128⁵⁵ [i]