MinT - Best Known (58−45, 58, s)-Nets in Base 64

Best Known (58−45, 58, s)-Nets in Base 64

(58−45, 58, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 58, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 58, 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]

(58−45, 58, 192)-Net in Base 64 — Constructive

(13, 58, 192)-net in base 64, using

12 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
- base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(58−45, 58, 257)-Net over F₆₄ — Digital

Digital (13, 58, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 58, 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]

(58−45, 58, 6862)-Net in Base 64 — Upper bound on s

There is no (13, 58, 6863)-net in base 64, because

1 times m-reduction [i] would yield (13, 57, 6863)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8 976782 197549 442552 528983 013245 645461 683709 620621 725527 092404 983379 439796 854531 747637 338646 710062 123588 > 64⁵⁷ [i]