MinT - Best Known (51, 51+56, s)-Nets in Base 8

Best Known (51, 51+56, s)-Nets in Base 8

(51, 51+56, 110)-Net over F₈ — Constructive and digital

Digital (51, 107, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 37, 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, 70, 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]

(51, 51+56, 153)-Net over F₈ — Digital

Digital (51, 107, 153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+56, 4542)-Net in Base 8 — Upper bound on s

There is no (51, 107, 4543)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 274625 863891 280942 519017 625054 861426 713401 078207 604634 618896 074043 154954 343064 793535 597959 435985 > 8¹⁰⁷ [i]