MinT - Best Known (57, 57+69, s)-Nets in Base 8

Best Known (57, 57+69, s)-Nets in Base 8

(57, 57+69, 110)-Net over F₈ — Constructive and digital

Digital (57, 126, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 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, 83, 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]

(57, 57+69, 146)-Net over F₈ — Digital

Digital (57, 126, 146)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+69, 4020)-Net in Base 8 — Upper bound on s

There is no (57, 126, 4021)-net in base 8, because

1 times m-reduction [i] would yield (57, 125, 4021)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 77378 789288 813148 141891 783037 283879 074363 787414 632364 248485 857953 787934 182495 846960 756358 044009 522083 027177 206618 > 8¹²⁵ [i]