MinT - Best Known (139−75, 139, s)-Nets in Base 8

Best Known (139−75, 139, s)-Nets in Base 8

(139−75, 139, 113)-Net over F₈ — Constructive and digital

Digital (64, 139, 113)-net over F₈, using

3 times m-reduction [i] based on digital (64, 142, 113)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 50, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. digital (14, 92, 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]

(139−75, 139, 169)-Net over F₈ — Digital

Digital (64, 139, 169)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−75, 139, 4864)-Net in Base 8 — Upper bound on s

There is no (64, 139, 4865)-net in base 8, because

1 times m-reduction [i] would yield (64, 138, 4865)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42518 008100 443347 648378 623573 586350 065616 328031 163166 524759 393462 031838 599709 746590 053157 856232 306686 427897 526315 933914 651392 > 8¹³⁸ [i]