MinT - Best Known (164−79, 164, s)-Nets in Base 8

Best Known (164−79, 164, s)-Nets in Base 8

(164−79, 164, 208)-Net over F₈ — Constructive and digital

Digital (85, 164, 208)-net over F₈, using

trace code for nets [i] based on digital (3, 82, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(164−79, 164, 225)-Net in Base 8 — Constructive

(85, 164, 225)-net in base 8, using

t-expansion [i] based on (83, 164, 225)-net in base 8, using
- 8 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(164−79, 164, 307)-Net over F₈ — Digital

Digital (85, 164, 307)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−79, 164, 13060)-Net in Base 8 — Upper bound on s

There is no (85, 164, 13061)-net in base 8, because

1 times m-reduction [i] would yield (85, 163, 13061)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1600 056326 241009 773526 492861 832229 150589 144773 881247 416328 079410 323011 318568 560153 823110 770350 960201 568750 885416 363117 474525 516564 383947 568562 557184 > 8¹⁶³ [i]