MinT - Best Known (158−59, 158, s)-Nets in Base 8

Best Known (158−59, 158, s)-Nets in Base 8

(158−59, 158, 354)-Net over F₈ — Constructive and digital

Digital (99, 158, 354)-net over F₈, using

t-expansion [i] based on digital (93, 158, 354)-net over F₈, using
- 14 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]

(158−59, 158, 514)-Net in Base 8 — Constructive

(99, 158, 514)-net in base 8, using

trace code for nets [i] based on (20, 79, 257)-net in base 64, using
- 1 times m-reduction [i] based on (20, 80, 257)-net in base 64, using
  - base change [i] based on digital (0, 60, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(158−59, 158, 937)-Net over F₈ — Digital

Digital (99, 158, 937)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−59, 158, 129153)-Net in Base 8 — Upper bound on s

There is no (99, 158, 129154)-net in base 8, because

1 times m-reduction [i] would yield (99, 157, 129154)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6098 195259 999493 968185 581744 354700 771792 110982 558509 584997 872936 895365 792682 576043 491436 982165 453357 634443 069050 732047 596329 396143 685573 132640 > 8¹⁵⁷ [i]