MinT - Best Known (12, 12+29, s)-Nets in Base 64

Best Known (12, 12+29, s)-Nets in Base 64

(12, 12+29, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 41, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 41, 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]

(12, 12+29, 257)-Net over F₆₄ — Digital

Digital (12, 41, 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]

(12, 12+29, 258)-Net in Base 64 — Constructive

(12, 41, 258)-net in base 64, using

3 times m-reduction [i] based on (12, 44, 258)-net in base 64, using
- base change [i] based on digital (1, 33, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(12, 12+29, 289)-Net in Base 64

(12, 41, 289)-net in base 64, using

3 times m-reduction [i] based on (12, 44, 289)-net in base 64, using
- base change [i] based on digital (1, 33, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(12, 12+29, 13881)-Net in Base 64 — Upper bound on s

There is no (12, 41, 13882)-net in base 64, because

1 times m-reduction [i] would yield (12, 40, 13882)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 768606 130186 762265 243508 694604 814975 789946 232346 351079 509213 433031 892360 > 64⁴⁰ [i]