MinT - Best Known (40, 40+38, s)-Nets in Base 8

Best Known (40, 40+38, s)-Nets in Base 8

(40, 40+38, 160)-Net over F₈ — Constructive and digital

Digital (40, 78, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 39, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(40, 40+38, 162)-Net over F₈ — Digital

Digital (40, 78, 162)-net over F₈, using

trace code for nets [i] based on digital (1, 39, 81)-net over F₆₄, using
- net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(40, 40+38, 5763)-Net in Base 8 — Upper bound on s

There is no (40, 78, 5764)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27686 359677 967763 186102 252884 593190 211864 143668 667519 365623 746160 450728 > 8⁷⁸ [i]