MinT - Best Known (164−61, 164, s)-Nets in Base 8

Best Known (164−61, 164, s)-Nets in Base 8

(164−61, 164, 354)-Net over F₈ — Constructive and digital

Digital (103, 164, 354)-net over F₈, using

t-expansion [i] based on digital (93, 164, 354)-net over F₈, using
- 8 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]

(164−61, 164, 576)-Net in Base 8 — Constructive

(103, 164, 576)-net in base 8, using

trace code for nets [i] based on (21, 82, 288)-net in base 64, using
- 2 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(164−61, 164, 986)-Net over F₈ — Digital

Digital (103, 164, 986)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−61, 164, 138810)-Net in Base 8 — Upper bound on s

There is no (103, 164, 138811)-net in base 8, because

1 times m-reduction [i] would yield (103, 163, 138811)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 676723 668984 319513 053326 834911 934520 307996 314768 732728 653199 383781 498075 225210 309329 611794 485617 153447 412818 683490 846708 409771 619219 567515 798984 > 8¹⁶³ [i]