MinT - Best Known (38−25, 38, s)-Nets in Base 64

Best Known (38−25, 38, s)-Nets in Base 64

(38−25, 38, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 38, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 38, 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]

(38−25, 38, 257)-Net over F₆₄ — Digital

Digital (13, 38, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 38, 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]

(38−25, 38, 260)-Net in Base 64 — Constructive

(13, 38, 260)-net in base 64, using

2 times m-reduction [i] based on (13, 40, 260)-net in base 64, using
- base change [i] based on digital (3, 30, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−25, 38, 321)-Net in Base 64

(13, 38, 321)-net in base 64, using

6 times m-reduction [i] based on (13, 44, 321)-net in base 64, using
- base change [i] based on digital (2, 33, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(38−25, 38, 31116)-Net in Base 64 — Upper bound on s

There is no (13, 38, 31117)-net in base 64, because

1 times m-reduction [i] would yield (13, 37, 31117)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 6 740237 419703 769914 181122 947136 096405 111149 944786 504564 702960 050545 > 64³⁷ [i]