MinT - Best Known (170−67, 170, s)-Nets in Base 8

Best Known (170−67, 170, s)-Nets in Base 8

(170−67, 170, 354)-Net over F₈ — Constructive and digital

Digital (103, 170, 354)-net over F₈, using

t-expansion [i] based on digital (93, 170, 354)-net over F₈, using
- 2 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]

(170−67, 170, 432)-Net in Base 8 — Constructive

(103, 170, 432)-net in base 8, using

8² times duplication [i] based on (101, 168, 432)-net in base 8, using
- trace code for nets [i] based on (17, 84, 216)-net in base 64, using
  - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(170−67, 170, 763)-Net over F₈ — Digital

Digital (103, 170, 763)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−67, 170, 79260)-Net in Base 8 — Upper bound on s

There is no (103, 170, 79261)-net in base 8, because

1 times m-reduction [i] would yield (103, 169, 79261)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 043777 756059 656423 260120 714785 278221 215413 760291 422849 808618 578165 092126 468585 891749 214114 236611 530554 805559 771351 196636 817776 971894 485122 786568 847632 > 8¹⁶⁹ [i]