MinT - Best Known (141−57, 141, s)-Nets in Base 8

Best Known (141−57, 141, s)-Nets in Base 8

(141−57, 141, 354)-Net over F₈ — Constructive and digital

Digital (84, 141, 354)-net over F₈, using

13 times m-reduction [i] based on digital (84, 154, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 77, 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]

(141−57, 141, 384)-Net in Base 8 — Constructive

(84, 141, 384)-net in base 8, using

8¹ times duplication [i] based on (83, 140, 384)-net in base 8, using
- trace code for nets [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(141−57, 141, 578)-Net over F₈ — Digital

Digital (84, 141, 578)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(141−57, 141, 52870)-Net in Base 8 — Upper bound on s

There is no (84, 141, 52871)-net in base 8, because

1 times m-reduction [i] would yield (84, 140, 52871)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 708437 922562 869322 100463 764689 588395 283200 151189 322228 448106 233011 163957 953613 717072 449298 881708 988959 570576 593736 360734 778512 > 8¹⁴⁰ [i]