MinT - Best Known (165−67, 165, s)-Nets in Base 8

Best Known (165−67, 165, s)-Nets in Base 8

(165−67, 165, 354)-Net over F₈ — Constructive and digital

Digital (98, 165, 354)-net over F₈, using

t-expansion [i] based on digital (93, 165, 354)-net over F₈, using
- 7 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(165−67, 165, 384)-Net in Base 8 — Constructive

(98, 165, 384)-net in base 8, using

1 times m-reduction [i] based on (98, 166, 384)-net in base 8, using
- trace code for nets [i] based on (15, 83, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
    - base change [i] based on digital (3, 72, 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]

(165−67, 165, 646)-Net over F₈ — Digital

Digital (98, 165, 646)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−67, 165, 57834)-Net in Base 8 — Upper bound on s

There is no (98, 165, 57835)-net in base 8, because

1 times m-reduction [i] would yield (98, 164, 57835)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12793 095505 379704 224285 561166 881856 014743 157933 487245 980578 956100 687264 495395 655786 122454 575350 328501 375135 517975 595947 667241 257495 803293 260143 484450 > 8¹⁶⁴ [i]