MinT - Best Known (171−65, 171, s)-Nets in Base 8

Best Known (171−65, 171, s)-Nets in Base 8

(171−65, 171, 354)-Net over F₈ — Constructive and digital

Digital (106, 171, 354)-net over F₈, using

t-expansion [i] based on digital (93, 171, 354)-net over F₈, using
- 1 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]

(171−65, 171, 432)-Net in Base 8 — Constructive

(106, 171, 432)-net in base 8, using

t-expansion [i] based on (104, 171, 432)-net in base 8, using
- 1 times m-reduction [i] based on (104, 172, 432)-net in base 8, using
  - trace code for nets [i] based on (18, 86, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 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]

(171−65, 171, 915)-Net over F₈ — Digital

Digital (106, 171, 915)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−65, 171, 114649)-Net in Base 8 — Upper bound on s

There is no (106, 171, 114650)-net in base 8, because

1 times m-reduction [i] would yield (106, 170, 114650)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3352 494389 978780 729363 455970 540146 835913 681880 813007 287203 647989 566518 435370 465150 176705 332883 202170 521717 001528 634049 510691 966222 957524 933583 532423 197238 > 8¹⁷⁰ [i]