MinT - Best Known (83−61, 83, s)-Nets in Base 64

Best Known (83−61, 83, s)-Nets in Base 64

(83−61, 83, 177)-Net over F₆₄ — Constructive and digital

Digital (22, 83, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 83, 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]

(83−61, 83, 288)-Net in Base 64 — Constructive

(22, 83, 288)-net in base 64, using

8 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 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]

(83−61, 83, 342)-Net over F₆₄ — Digital

Digital (22, 83, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 83, 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]

(83−61, 83, 16517)-Net in Base 64 — Upper bound on s

There is no (22, 83, 16518)-net in base 64, because

1 times m-reduction [i] would yield (22, 82, 16518)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 12796 331970 690526 896347 865377 281869 379648 063236 377156 302987 060257 807609 110888 957865 735098 128716 625431 069200 066785 835758 414153 354209 031168 913424 584512 > 64⁸² [i]