MinT - Best Known (87, 87+83, s)-Nets in Base 8

Best Known (87, 87+83, s)-Nets in Base 8

(87, 87+83, 194)-Net over F₈ — Constructive and digital

Digital (87, 170, 194)-net over F₈, using

t-expansion [i] based on digital (85, 170, 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]

(87, 87+83, 225)-Net in Base 8 — Constructive

(87, 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]

(87, 87+83, 299)-Net over F₈ — Digital

Digital (87, 170, 299)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+83, 12144)-Net in Base 8 — Upper bound on s

There is no (87, 170, 12145)-net in base 8, because

1 times m-reduction [i] would yield (87, 169, 12145)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 501955 622502 288353 715213 460410 065445 684401 199615 143622 100258 215467 818285 309374 577648 554324 271718 170523 268474 364951 171130 523685 339981 100581 129520 078080 > 8¹⁶⁹ [i]