MinT - Best Known (157−85, 157, s)-Nets in Base 8

Best Known (157−85, 157, s)-Nets in Base 8

(157−85, 157, 130)-Net over F₈ — Constructive and digital

Digital (72, 157, 130)-net over F₈, using

3 times m-reduction [i] based on digital (72, 160, 130)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 58, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (14, 102, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(157−85, 157, 184)-Net over F₈ — Digital

Digital (72, 157, 184)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−85, 157, 5307)-Net in Base 8 — Upper bound on s

There is no (72, 157, 5308)-net in base 8, because

1 times m-reduction [i] would yield (72, 156, 5308)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 766 017656 749250 137844 435714 760212 879188 361837 669400 767673 606869 590914 640490 462401 862228 497497 549553 855804 185568 779581 961951 661953 522980 415759 > 8¹⁵⁶ [i]