MinT - Best Known (170−86, 170, s)-Nets in Base 8

Best Known (170−86, 170, s)-Nets in Base 8

(170−86, 170, 130)-Net over F₈ — Constructive and digital

Digital (84, 170, 130)-net over F₈, using

t-expansion [i] based on digital (76, 170, 130)-net over F₈, using
- 2 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 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, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(170−86, 170, 225)-Net in Base 8 — Constructive

(84, 170, 225)-net in base 8, using

t-expansion [i] based on (83, 170, 225)-net in base 8, using
- 2 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]

(170−86, 170, 259)-Net over F₈ — Digital

Digital (84, 170, 259)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−86, 170, 8941)-Net in Base 8 — Upper bound on s

There is no (84, 170, 8942)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3359 754122 383225 678926 572898 630627 678815 635585 083725 479492 762000 587349 968411 145770 438067 829895 694961 161604 915558 174312 481605 064169 787878 383306 965950 125990 > 8¹⁷⁰ [i]