MinT - Best Known (81−61, 81, s)-Nets in Base 64

Best Known (81−61, 81, s)-Nets in Base 64

(81−61, 81, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 81, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 81, 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]

(81−61, 81, 216)-Net in Base 64 — Constructive

(20, 81, 216)-net in base 64, using

t-expansion [i] based on (18, 81, 216)-net in base 64, using
- 10 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]

(81−61, 81, 342)-Net over F₆₄ — Digital

Digital (20, 81, 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]

(81−61, 81, 12514)-Net in Base 64 — Upper bound on s

There is no (20, 81, 12515)-net in base 64, because

1 times m-reduction [i] would yield (20, 80, 12515)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 126511 935214 463025 026367 058631 591908 616877 953155 403570 766347 306657 449727 343841 423845 659752 809514 195224 725007 383126 885308 591443 545826 373065 590200 > 64⁸⁰ [i]