MinT - Best Known (164−78, 164, s)-Nets in Base 8

Best Known (164−78, 164, s)-Nets in Base 8

(164−78, 164, 208)-Net over F₈ — Constructive and digital

Digital (86, 164, 208)-net over F₈, using

2 times m-reduction [i] based on digital (86, 166, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 83, 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]

(164−78, 164, 225)-Net in Base 8 — Constructive

(86, 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]

(164−78, 164, 324)-Net over F₈ — Digital

Digital (86, 164, 324)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−78, 164, 13777)-Net in Base 8 — Upper bound on s

There is no (86, 164, 13778)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12813 336697 683798 256595 938975 319044 637009 499586 531132 177553 005628 510620 926512 646054 777627 877522 736097 020449 439572 513559 176176 975652 200756 518689 497040 > 8¹⁶⁴ [i]