MinT - Best Known (168−51, 168, s)-Nets in Base 8

Best Known (168−51, 168, s)-Nets in Base 8

(168−51, 168, 1026)-Net over F₈ — Constructive and digital

Digital (117, 168, 1026)-net over F₈, using

t-expansion [i] based on digital (114, 168, 1026)-net over F₈, using
- 4 times m-reduction [i] based on digital (114, 172, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(168−51, 168, 3038)-Net over F₈ — Digital

Digital (117, 168, 3038)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−51, 168, 1567344)-Net in Base 8 — Upper bound on s

There is no (117, 168, 1567345)-net in base 8, because

1 times m-reduction [i] would yield (117, 167, 1567345)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 546787 692261 189583 780075 680600 984801 023708 795232 753742 968882 218218 425995 617378 733247 035285 616547 998825 442570 913120 852639 744522 474011 403610 447928 598464 > 8¹⁶⁷ [i]