MinT - Best Known (172−87, 172, s)-Nets in Base 8

Best Known (172−87, 172, s)-Nets in Base 8

(172−87, 172, 194)-Net over F₈ — Constructive and digital

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]

(172−87, 172, 225)-Net in Base 8 — Constructive

(85, 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]

(172−87, 172, 262)-Net over F₈ — Digital

Digital (85, 172, 262)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−87, 172, 9385)-Net in Base 8 — Upper bound on s

There is no (85, 172, 9386)-net in base 8, because

1 times m-reduction [i] would yield (85, 171, 9386)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26829 448550 286549 592424 990000 226947 522148 803008 073971 396174 788229 795327 420871 083639 693162 375987 815628 354183 602851 670920 674717 684697 597355 750819 903446 294016 > 8¹⁷¹ [i]