MinT - Best Known (54−41, 54, s)-Nets in Base 64

Best Known (54−41, 54, s)-Nets in Base 64

(54−41, 54, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 54, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 54, 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]

(54−41, 54, 216)-Net in Base 64 — Constructive

(13, 54, 216)-net in base 64, using

2 times m-reduction [i] based on (13, 56, 216)-net in base 64, using
- base change [i] based on digital (5, 48, 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]

(54−41, 54, 257)-Net over F₆₄ — Digital

Digital (13, 54, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 54, 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]

(54−41, 54, 8050)-Net in Base 64 — Upper bound on s

There is no (13, 54, 8051)-net in base 64, because

1 times m-reduction [i] would yield (13, 53, 8051)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 534891 222492 874850 662112 092633 576279 881063 616074 238812 615395 742421 305037 353124 265354 201072 270536 > 64⁵³ [i]