MinT - Best Known (140−75, 140, s)-Nets in Base 8

Best Known (140−75, 140, s)-Nets in Base 8

(140−75, 140, 130)-Net over F₈ — Constructive and digital

Digital (65, 140, 130)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 51, 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, 89, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(140−75, 140, 175)-Net over F₈ — Digital

Digital (65, 140, 175)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−75, 140, 5146)-Net in Base 8 — Upper bound on s

There is no (65, 140, 5147)-net in base 8, because

1 times m-reduction [i] would yield (65, 139, 5147)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338710 975297 090921 261637 004481 918249 074899 017480 970988 650079 812095 049256 634976 342973 838235 535579 509898 238098 507436 469761 888128 > 8¹³⁹ [i]