MinT - Best Known (156−59, 156, s)-Nets in Base 8

Best Known (156−59, 156, s)-Nets in Base 8

(156−59, 156, 354)-Net over F₈ — Constructive and digital

Digital (97, 156, 354)-net over F₈, using

t-expansion [i] based on digital (93, 156, 354)-net over F₈, using
- 16 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]

(156−59, 156, 432)-Net in Base 8 — Constructive

(97, 156, 432)-net in base 8, using

4 times m-reduction [i] based on (97, 160, 432)-net in base 8, using
- trace code for nets [i] based on (17, 80, 216)-net in base 64, using
  - 4 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(156−59, 156, 870)-Net over F₈ — Digital

Digital (97, 156, 870)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−59, 156, 111895)-Net in Base 8 — Upper bound on s

There is no (97, 156, 111896)-net in base 8, because

1 times m-reduction [i] would yield (97, 155, 111896)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 272694 986837 267526 824965 205146 397229 508521 728935 332246 281196 958334 869380 854386 546749 079483 716009 583068 110928 764189 457487 547290 298528 871328 > 8¹⁵⁵ [i]