MinT - Best Known (171−85, 171, s)-Nets in Base 8

Best Known (171−85, 171, s)-Nets in Base 8

(171−85, 171, 194)-Net over F₈ — Constructive and digital

Digital (86, 171, 194)-net over F₈, using

t-expansion [i] based on digital (85, 171, 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]

(171−85, 171, 225)-Net in Base 8 — Constructive

(86, 171, 225)-net in base 8, using

t-expansion [i] based on (83, 171, 225)-net in base 8, using
- 1 times m-reduction [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]

(171−85, 171, 280)-Net over F₈ — Digital

Digital (86, 171, 280)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−85, 171, 10641)-Net in Base 8 — Upper bound on s

There is no (86, 171, 10642)-net in base 8, because

1 times m-reduction [i] would yield (86, 170, 10642)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3363 151147 499530 867886 216925 292974 378268 360806 345871 998699 889764 472136 242393 364178 439435 835301 720666 835705 675439 858186 420272 331516 627237 932467 502675 448160 > 8¹⁷⁰ [i]