MinT - Best Known (154−61, 154, s)-Nets in Base 8

Best Known (154−61, 154, s)-Nets in Base 8

(154−61, 154, 354)-Net over F₈ — Constructive and digital

Digital (93, 154, 354)-net over F₈, using

18 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]

(154−61, 154, 432)-Net in Base 8 — Constructive

(93, 154, 432)-net in base 8, using

trace code for nets [i] based on (16, 77, 216)-net in base 64, using
- base change [i] based on digital (5, 66, 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]

(154−61, 154, 685)-Net over F₈ — Digital

Digital (93, 154, 685)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−61, 154, 69395)-Net in Base 8 — Upper bound on s

There is no (93, 154, 69396)-net in base 8, because

1 times m-reduction [i] would yield (93, 153, 69396)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488814 026361 609772 600011 836033 249351 750284 571374 681207 666090 926542 162606 850476 486731 112566 853500 723073 404495 859488 351929 735912 693564 899648 > 8¹⁵³ [i]