MinT - Best Known (170−63, 170, s)-Nets in Base 8

Best Known (170−63, 170, s)-Nets in Base 8

(170−63, 170, 354)-Net over F₈ — Constructive and digital

Digital (107, 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]

(170−63, 170, 576)-Net in Base 8 — Constructive

(107, 170, 576)-net in base 8, using

8² times duplication [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [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]

(170−63, 170, 1035)-Net over F₈ — Digital

Digital (107, 170, 1035)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−63, 170, 148653)-Net in Base 8 — Upper bound on s

There is no (107, 170, 148654)-net in base 8, because

1 times m-reduction [i] would yield (107, 169, 148654)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 036123 418259 802743 989789 129122 658497 001051 542952 487245 225762 112911 761353 727555 026059 508411 492411 328651 069304 501563 867732 458952 480134 401683 396493 397520 > 8¹⁶⁹ [i]