MinT - Best Known (172−86, 172, s)-Nets in Base 8

Best Known (172−86, 172, s)-Nets in Base 8

(172−86, 172, 194)-Net over F₈ — Constructive and digital

Digital (86, 172, 194)-net over F₈, using

t-expansion [i] based on digital (85, 172, 194)-net over F₈, using
- net from sequence [i] based on digital (85, 193)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 85 and N(F) ≥ 194, using
    - F₆ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(172−86, 172, 225)-Net in Base 8 — Constructive

(86, 172, 225)-net in base 8, using

t-expansion [i] based on (83, 172, 225)-net in base 8, using
- base change [i] based on digital (40, 129, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(172−86, 172, 274)-Net over F₈ — Digital

Digital (86, 172, 274)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−86, 172, 9852)-Net in Base 8 — Upper bound on s

There is no (86, 172, 9853)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 215201 256044 675359 977498 991177 044221 595543 264222 401166 982282 686044 630821 603398 204476 805290 590189 775001 587487 458315 730271 147882 482487 073370 903360 864464 958252 > 8¹⁷² [i]