MinT - Best Known (68−50, 68, s)-Nets in Base 64

Best Known (68−50, 68, s)-Nets in Base 64

(68−50, 68, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 68, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 68, 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]

(68−50, 68, 258)-Net in Base 64 — Constructive

(18, 68, 258)-net in base 64, using

base change [i] based on digital (1, 51, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(68−50, 68, 281)-Net over F₆₄ — Digital

Digital (18, 68, 281)-net over F₆₄, using

net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
  - a curve from the manYPoints database [i]

(68−50, 68, 289)-Net in Base 64

(18, 68, 289)-net in base 64, using

base change [i] based on digital (1, 51, 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]

(68−50, 68, 13203)-Net in Base 64 — Upper bound on s

There is no (18, 68, 13204)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 661 748384 899780 240179 764874 306959 537719 629686 944294 973081 144976 536889 384950 753806 196884 883775 590230 943924 589299 248585 756528 > 64⁶⁸ [i]