MinT - Best Known (157−63, 157, s)-Nets in Base 8

Best Known (157−63, 157, s)-Nets in Base 8

(157−63, 157, 354)-Net over F₈ — Constructive and digital

Digital (94, 157, 354)-net over F₈, using

t-expansion [i] based on digital (93, 157, 354)-net over F₈, using
- 15 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(157−63, 157, 384)-Net in Base 8 — Constructive

(94, 157, 384)-net in base 8, using

1 times m-reduction [i] based on (94, 158, 384)-net in base 8, using
- trace code for nets [i] based on (15, 79, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
    - base change [i] based on digital (3, 72, 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]

(157−63, 157, 655)-Net over F₈ — Digital

Digital (94, 157, 655)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−63, 157, 62141)-Net in Base 8 — Upper bound on s

There is no (94, 157, 62142)-net in base 8, because

1 times m-reduction [i] would yield (94, 156, 62142)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 762 373261 018075 295123 433406 901870 736244 999332 077304 873561 469133 555435 894575 453740 456214 491508 555089 153690 003598 983355 684079 980401 046998 705840 > 8¹⁵⁶ [i]