MinT - Best Known (66, 66+31, s)-Nets in Base 8

Best Known (66, 66+31, s)-Nets in Base 8

(66, 66+31, 382)-Net over F₈ — Constructive and digital

Digital (66, 97, 382)-net over F₈, using

8¹ times duplication [i] based on digital (65, 96, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 20, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (45, 76, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 38, 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]

(66, 66+31, 576)-Net in Base 8 — Constructive

(66, 97, 576)-net in base 8, using

t-expansion [i] based on (65, 97, 576)-net in base 8, using
- 1 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(66, 66+31, 1446)-Net over F₈ — Digital

Digital (66, 97, 1446)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+31, 552633)-Net in Base 8 — Upper bound on s

There is no (66, 97, 552634)-net in base 8, because

1 times m-reduction [i] would yield (66, 96, 552634)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 497 327843 191461 047999 731746 958466 253578 138957 188798 670108 744255 679637 335133 874168 186776 > 8⁹⁶ [i]