MinT - Best Known (167−61, 167, s)-Nets in Base 8

Best Known (167−61, 167, s)-Nets in Base 8

(167−61, 167, 368)-Net over F₈ — Constructive and digital

Digital (106, 167, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 31, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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]

(167−61, 167, 576)-Net in Base 8 — Constructive

(106, 167, 576)-net in base 8, using

t-expansion [i] based on (105, 167, 576)-net in base 8, using
- 1 times m-reduction [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]

(167−61, 167, 1100)-Net over F₈ — Digital

Digital (106, 167, 1100)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−61, 167, 170899)-Net in Base 8 — Upper bound on s

There is no (106, 167, 170900)-net in base 8, because

1 times m-reduction [i] would yield (106, 166, 170900)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 818417 748310 466099 872031 008830 837591 541995 131006 478572 937202 452313 373995 490436 243124 105677 450823 075568 473825 480658 872313 990743 129653 725567 598511 024288 > 8¹⁶⁶ [i]