MinT - Best Known (162−65, 162, s)-Nets in Base 8

Best Known (162−65, 162, s)-Nets in Base 8

(162−65, 162, 354)-Net over F₈ — Constructive and digital

Digital (97, 162, 354)-net over F₈, using

t-expansion [i] based on digital (93, 162, 354)-net over F₈, using
- 10 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]

(162−65, 162, 384)-Net in Base 8 — Constructive

(97, 162, 384)-net in base 8, using

2 times m-reduction [i] based on (97, 164, 384)-net in base 8, using
- trace code for nets [i] based on (15, 82, 192)-net in base 64, using
  - 2 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]

(162−65, 162, 672)-Net over F₈ — Digital

Digital (97, 162, 672)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−65, 162, 63873)-Net in Base 8 — Upper bound on s

There is no (97, 162, 63874)-net in base 8, because

1 times m-reduction [i] would yield (97, 161, 63874)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 986367 673461 059059 538655 605169 899538 816128 030230 163967 832361 363581 596536 026759 535764 502903 228987 763733 514431 053064 080340 537732 836719 872881 103572 > 8¹⁶¹ [i]