MinT - Best Known (79−37, 79, s)-Nets in Base 8

Best Known (79−37, 79, s)-Nets in Base 8

(79−37, 79, 160)-Net over F₈ — Constructive and digital

Digital (42, 79, 160)-net over F₈, using

3 times m-reduction [i] based on digital (42, 82, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 41, 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]

(79−37, 79, 194)-Net over F₈ — Digital

Digital (42, 79, 194)-net over F₈, using

1 times m-reduction [i] based on digital (42, 80, 194)-net over F₈, using
- trace code for nets [i] based on digital (2, 40, 97)-net over F₆₄, using
  - net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(79−37, 79, 8828)-Net in Base 8 — Upper bound on s

There is no (42, 79, 8829)-net in base 8, because

1 times m-reduction [i] would yield (42, 78, 8829)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27645 458164 113970 903238 774108 142018 195968 515347 206116 760376 565136 782690 > 8⁷⁸ [i]