MinT - Best Known (25, 72, s)-Nets in Base 64

Best Known (25, 72, s)-Nets in Base 64

(25, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (25, 72, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 72, 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]

(25, 72, 288)-Net in Base 64 — Constructive

(25, 72, 288)-net in base 64, using

t-expansion [i] based on (22, 72, 288)-net in base 64, using
- 19 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]

(25, 72, 408)-Net over F₆₄ — Digital

Digital (25, 72, 408)-net over F₆₄, using

net from sequence [i] based on digital (25, 407)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 25 and N(F) ≥ 408, using
  - a curve from the manYPoints database [i]

(25, 72, 56315)-Net in Base 64 — Upper bound on s

There is no (25, 72, 56316)-net in base 64, because

1 times m-reduction [i] would yield (25, 71, 56316)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 173 334556 837101 856478 540379 550555 629405 136115 992534 636337 639456 862462 979749 469128 622751 604679 542664 039959 548653 940028 173060 874766 > 64⁷¹ [i]