MinT - Best Known (86−65, 86, s)-Nets in Base 64

Best Known (86−65, 86, s)-Nets in Base 64

(86−65, 86, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 86, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 86, 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]

(86−65, 86, 216)-Net in Base 64 — Constructive

(21, 86, 216)-net in base 64, using

t-expansion [i] based on (18, 86, 216)-net in base 64, using
- 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
  - base change [i] based on digital (5, 78, 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]

(86−65, 86, 342)-Net over F₆₄ — Digital

Digital (21, 86, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 86, 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]

(86−65, 86, 12725)-Net in Base 64 — Upper bound on s

There is no (21, 86, 12726)-net in base 64, because

1 times m-reduction [i] would yield (21, 85, 12726)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3359 954396 167498 987396 444654 632161 507419 767077 383811 540759 862609 240288 151665 995868 530746 235220 049301 079710 976780 435170 487865 236793 482984 891698 050307 311939 > 64⁸⁵ [i]