MinT - Best Known (90, 90+59, s)-Nets in Base 8

Best Known (90, 90+59, s)-Nets in Base 8

(90, 90+59, 354)-Net over F₈ — Constructive and digital

Digital (90, 149, 354)-net over F₈, using

17 times m-reduction [i] based on digital (90, 166, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 83, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(90, 90+59, 384)-Net in Base 8 — Constructive

(90, 149, 384)-net in base 8, using

3 times m-reduction [i] based on (90, 152, 384)-net in base 8, using
- trace code for nets [i] based on (14, 76, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(90, 90+59, 668)-Net over F₈ — Digital

Digital (90, 149, 668)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 90+59, 67729)-Net in Base 8 — Upper bound on s

There is no (90, 149, 67730)-net in base 8, because

1 times m-reduction [i] would yield (90, 148, 67730)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 45 428511 456253 601490 446049 470272 716414 274422 543272 264006 233674 279248 298143 856281 570389 496373 652999 512807 679686 697621 037219 796433 441696 > 8¹⁴⁸ [i]