MinT - Best Known (59, 59+31, s)-Nets in Base 8

Best Known (59, 59+31, s)-Nets in Base 8

(59, 59+31, 354)-Net over F₈ — Constructive and digital

Digital (59, 90, 354)-net over F₈, using

14 times m-reduction [i] based on digital (59, 104, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 52, 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]

(59, 59+31, 518)-Net in Base 8 — Constructive

(59, 90, 518)-net in base 8, using

8² times duplication [i] based on (57, 88, 518)-net in base 8, using
- base change [i] based on digital (35, 66, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 33, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(59, 59+31, 896)-Net over F₈ — Digital

Digital (59, 90, 896)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+31, 209403)-Net in Base 8 — Upper bound on s

There is no (59, 90, 209404)-net in base 8, because

1 times m-reduction [i] would yield (59, 89, 209404)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 153755 114940 834902 841328 284846 812914 116017 106943 635066 786835 724508 453646 932772 > 8⁸⁹ [i]