MinT - Best Known (157−59, 157, s)-Nets in Base 8

Best Known (157−59, 157, s)-Nets in Base 8

(157−59, 157, 354)-Net over F₈ — Constructive and digital

Digital (98, 157, 354)-net over F₈, using

t-expansion [i] based on digital (93, 157, 354)-net over F₈, using
- 15 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]

(157−59, 157, 432)-Net in Base 8 — Constructive

(98, 157, 432)-net in base 8, using

5 times m-reduction [i] based on (98, 162, 432)-net in base 8, using
- trace code for nets [i] based on (17, 81, 216)-net in base 64, using
  - 3 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]

(157−59, 157, 903)-Net over F₈ — Digital

Digital (98, 157, 903)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−59, 157, 120215)-Net in Base 8 — Upper bound on s

There is no (98, 157, 120216)-net in base 8, because

1 times m-reduction [i] would yield (98, 156, 120216)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 762 273444 350294 970224 073623 358160 890996 193067 929729 671667 017182 662826 765578 690717 193363 481419 944716 111218 996575 346399 710787 048745 895519 915184 > 8¹⁵⁶ [i]