MinT - Best Known (48, 48+51, s)-Nets in Base 8

Best Known (48, 48+51, s)-Nets in Base 8

(48, 48+51, 110)-Net over F₈ — Constructive and digital

Digital (48, 99, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (14, 65, 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]

(48, 48+51, 153)-Net over F₈ — Digital

Digital (48, 99, 153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(48, 48+51, 5026)-Net in Base 8 — Upper bound on s

There is no (48, 99, 5027)-net in base 8, because

1 times m-reduction [i] would yield (48, 98, 5027)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31847 461065 999583 621156 142767 099089 712935 138554 416792 994792 294503 121182 744310 687401 173000 > 8⁹⁸ [i]