MinT - Best Known (21, 40, s)-Nets in Base 8

Best Known (21, 40, s)-Nets in Base 8

(21, 40, 160)-Net over F₈ — Constructive and digital

Digital (21, 40, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 20, 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]

(21, 40, 162)-Net over F₈ — Digital

Digital (21, 40, 162)-net over F₈, using

trace code for nets [i] based on digital (1, 20, 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]

(21, 40, 4848)-Net in Base 8 — Upper bound on s

There is no (21, 40, 4849)-net in base 8, because

1 times m-reduction [i] would yield (21, 39, 4849)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 166435 440844 641806 343045 844852 118208 > 8³⁹ [i]