MinT - Best Known (61−47, 61, s)-Nets in Base 64

Best Known (61−47, 61, s)-Nets in Base 64

(61−47, 61, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 61, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 61, 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]

(61−47, 61, 216)-Net in Base 64 — Constructive

(14, 61, 216)-net in base 64, using

2 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
- base change [i] based on digital (5, 54, 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]

(61−47, 61, 257)-Net over F₆₄ — Digital

Digital (14, 61, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 61, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(61−47, 61, 7695)-Net in Base 64 — Upper bound on s

There is no (14, 61, 7696)-net in base 64, because

1 times m-reduction [i] would yield (14, 60, 7696)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 350554 019176 784607 371548 627542 093893 532672 162819 224139 963086 246766 490982 336151 233378 636761 659757 785433 730144 > 64⁶⁰ [i]