MinT - Best Known (164−87, 164, s)-Nets in Base 8

Best Known (164−87, 164, s)-Nets in Base 8

(164−87, 164, 130)-Net over F₈ — Constructive and digital

Digital (77, 164, 130)-net over F₈, using

t-expansion [i] based on digital (76, 164, 130)-net over F₈, using
- 8 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)

(164−87, 164, 208)-Net over F₈ — Digital

Digital (77, 164, 208)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−87, 164, 6365)-Net in Base 8 — Upper bound on s

There is no (77, 164, 6366)-net in base 8, because

1 times m-reduction [i] would yield (77, 163, 6366)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 402973 703087 961316 515645 361220 363395 877247 493122 447321 996478 639037 565457 094329 419853 082035 900794 281566 578126 860690 198115 778312 043860 018162 401460 > 8¹⁶³ [i]