MinT - Best Known (51, 51+29, s)-Nets in Base 8

Best Known (51, 51+29, s)-Nets in Base 8

(51, 51+29, 354)-Net over F₈ — Constructive and digital

Digital (51, 80, 354)-net over F₈, using

8 times m-reduction [i] based on digital (51, 88, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 44, 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]

(51, 51+29, 516)-Net in Base 8 — Constructive

(51, 80, 516)-net in base 8, using

base change [i] based on digital (31, 60, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(51, 51+29, 628)-Net over F₈ — Digital

Digital (51, 80, 628)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+29, 107729)-Net in Base 8 — Upper bound on s

There is no (51, 80, 107730)-net in base 8, because

1 times m-reduction [i] would yield (51, 79, 107730)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220880 380875 193575 477766 955546 375575 724667 178598 451081 880549 448842 560904 > 8⁷⁹ [i]