MinT - Best Known (172−68, 172, s)-Nets in Base 8

Best Known (172−68, 172, s)-Nets in Base 8

(172−68, 172, 354)-Net over F₈ — Constructive and digital

Digital (104, 172, 354)-net over F₈, using

t-expansion [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]

(172−68, 172, 432)-Net in Base 8 — Constructive

(104, 172, 432)-net in base 8, using

trace code for nets [i] based on (18, 86, 216)-net in base 64, using
- 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
  - base change [i] based on digital (5, 78, 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]

(172−68, 172, 759)-Net over F₈ — Digital

Digital (104, 172, 759)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−68, 172, 71584)-Net in Base 8 — Upper bound on s

There is no (104, 172, 71585)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214546 820619 420194 019888 244767 793934 035531 236816 210534 319459 989367 810429 147721 359682 620230 498530 242665 011102 197215 761940 563200 760189 927852 288808 361291 797120 > 8¹⁷² [i]