MinT - Best Known (75−51, 75, s)-Nets in Base 64

Best Known (75−51, 75, s)-Nets in Base 64

(75−51, 75, 177)-Net over F₆₄ — Constructive and digital

Digital (24, 75, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 75, 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]

(75−51, 75, 288)-Net in Base 64 — Constructive

(24, 75, 288)-net in base 64, using

t-expansion [i] based on (22, 75, 288)-net in base 64, using
- 16 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]

(75−51, 75, 342)-Net over F₆₄ — Digital

Digital (24, 75, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 75, 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]

(75−51, 75, 35844)-Net in Base 64 — Upper bound on s

There is no (24, 75, 35845)-net in base 64, because

1 times m-reduction [i] would yield (24, 74, 35845)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 45 429372 051854 451827 064138 343164 592784 544298 394008 986529 722218 869417 852435 340704 795411 971117 010196 054196 195396 896623 855840 115399 223808 > 64⁷⁴ [i]