MinT - Best Known (89, 89+53, s)-Nets in Base 8

Best Known (89, 89+53, s)-Nets in Base 8

(89, 89+53, 354)-Net over F₈ — Constructive and digital

Digital (89, 142, 354)-net over F₈, using

22 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 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]

(89, 89+53, 514)-Net in Base 8 — Constructive

(89, 142, 514)-net in base 8, using

trace code for nets [i] based on (18, 71, 257)-net in base 64, using
- 1 times m-reduction [i] based on (18, 72, 257)-net in base 64, using
  - base change [i] based on digital (0, 54, 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]

(89, 89+53, 862)-Net over F₈ — Digital

Digital (89, 142, 862)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+53, 119031)-Net in Base 8 — Upper bound on s

There is no (89, 142, 119032)-net in base 8, because

1 times m-reduction [i] would yield (89, 141, 119032)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 663317 728856 491386 745261 142583 706862 284294 340606 717655 044207 574702 724720 122489 586000 309418 816574 853363 136270 870938 258659 058700 > 8¹⁴¹ [i]