MinT - Best Known (62−29, 62, s)-Nets in Base 8

Best Known (62−29, 62, s)-Nets in Base 8

(62−29, 62, 160)-Net over F₈ — Constructive and digital

Digital (33, 62, 160)-net over F₈, using

2 times m-reduction [i] based on digital (33, 64, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 32, 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]

(62−29, 62, 194)-Net over F₈ — Digital

Digital (33, 62, 194)-net over F₈, using

trace code for nets [i] based on digital (2, 31, 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]

(62−29, 62, 7426)-Net in Base 8 — Upper bound on s

There is no (33, 62, 7427)-net in base 8, because

1 times m-reduction [i] would yield (33, 61, 7427)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12 282992 120836 894673 266768 330849 809272 932288 138171 406528 > 8⁶¹ [i]