MinT - Best Known (56−36, 56, s)-Nets in Base 64

Best Known (56−36, 56, s)-Nets in Base 64

(56−36, 56, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 56, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 56, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(56−36, 56, 288)-Net in Base 64 — Constructive

(20, 56, 288)-net in base 64, using

21 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
- base change [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]

(56−36, 56, 342)-Net over F₆₄ — Digital

Digital (20, 56, 342)-net over F₆₄, using

net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
  - a curve from the manYPoints database [i]

(56−36, 56, 49881)-Net in Base 64 — Upper bound on s

There is no (20, 56, 49882)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 140008 693820 627860 096935 880107 381925 372871 598998 578042 544170 701330 873377 884927 402657 113638 396151 980044 > 64⁵⁶ [i]