MinT - Best Known (71, 71+45, s)-Nets in Base 8

Best Known (71, 71+45, s)-Nets in Base 8

(71, 71+45, 354)-Net over F₈ — Constructive and digital

Digital (71, 116, 354)-net over F₈, using

12 times m-reduction [i] based on digital (71, 128, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 64, 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]

(71, 71+45, 384)-Net in Base 8 — Constructive

(71, 116, 384)-net in base 8, using

2 times m-reduction [i] based on (71, 118, 384)-net in base 8, using
- trace code for nets [i] based on (12, 59, 192)-net in base 64, using
  - 4 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 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]

(71, 71+45, 604)-Net over F₈ — Digital

Digital (71, 116, 604)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+45, 67977)-Net in Base 8 — Upper bound on s

There is no (71, 116, 67978)-net in base 8, because

1 times m-reduction [i] would yield (71, 115, 67978)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 682301 101075 829665 378118 251805 476090 273764 143937 566377 755598 039733 293989 338156 506003 740150 709787 383712 > 8¹¹⁵ [i]