MinT - Best Known (83, 83+49, s)-Nets in Base 8

Best Known (83, 83+49, s)-Nets in Base 8

(83, 83+49, 354)-Net over F₈ — Constructive and digital

Digital (83, 132, 354)-net over F₈, using

20 times m-reduction [i] based on digital (83, 152, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 76, 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]

(83, 83+49, 514)-Net in Base 8 — Constructive

(83, 132, 514)-net in base 8, using

base change [i] based on digital (50, 99, 514)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (50, 100, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 50, 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]

(83, 83+49, 837)-Net over F₈ — Digital

Digital (83, 132, 837)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+49, 119008)-Net in Base 8 — Upper bound on s

There is no (83, 132, 119009)-net in base 8, because

1 times m-reduction [i] would yield (83, 131, 119009)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 20177 621776 817896 759229 090094 792781 776538 660195 013026 984398 870177 549887 920866 464036 285370 968451 696078 634300 226512 680916 > 8¹³¹ [i]