MinT - Best Known (83, 170, s)-Nets in Base 8

Best Known (83, 170, s)-Nets in Base 8

(83, 170, 130)-Net over F₈ — Constructive and digital

Digital (83, 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)

(83, 170, 225)-Net in Base 8 — Constructive

(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]

(83, 170, 248)-Net over F₈ — Digital

Digital (83, 170, 248)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 170, 8518)-Net in Base 8 — Upper bound on s

There is no (83, 170, 8519)-net in base 8, because

1 times m-reduction [i] would yield (83, 169, 8519)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 420 850759 597781 426910 677390 733802 915864 585728 812257 331848 380747 364389 872035 367080 656585 583050 078356 488354 981651 533377 889339 017205 998100 315253 768512 032160 > 8¹⁶⁹ [i]