MinT - Best Known (68−53, 68, s)-Nets in Base 64

Best Known (68−53, 68, s)-Nets in Base 64

(68−53, 68, 177)-Net over F₆₄ — Constructive and digital

Digital (15, 68, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 68, 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]

(68−53, 68, 216)-Net in Base 64 — Constructive

(15, 68, 216)-net in base 64, using

2 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
- base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(68−53, 68, 258)-Net over F₆₄ — Digital

Digital (15, 68, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(68−53, 68, 7543)-Net in Base 64 — Upper bound on s

There is no (15, 68, 7544)-net in base 64, because

1 times m-reduction [i] would yield (15, 67, 7544)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 10 331971 363853 999384 466133 447436 512402 037577 293954 358958 553972 683215 677139 102201 393236 819904 005403 117014 310046 293585 742396 > 64⁶⁷ [i]