MinT - Best Known (48−34, 48, s)-Nets in Base 64

Best Known (48−34, 48, s)-Nets in Base 64

(48−34, 48, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 48, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 48, 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]

(48−34, 48, 257)-Net over F₆₄ — Digital

Digital (14, 48, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 48, 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]

(48−34, 48, 259)-Net in Base 64 — Constructive

(14, 48, 259)-net in base 64, using

base change [i] based on digital (2, 36, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(48−34, 48, 321)-Net in Base 64

(14, 48, 321)-net in base 64, using

base change [i] based on digital (2, 36, 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]

(48−34, 48, 14327)-Net in Base 64 — Upper bound on s

There is no (14, 48, 14328)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 497 892003 733864 583770 736597 018362 971839 618918 582856 636014 437352 220073 857044 363993 098087 > 64⁴⁸ [i]