MinT - Best Known (52, 52+56, s)-Nets in Base 8

Best Known (52, 52+56, s)-Nets in Base 8

(52, 52+56, 111)-Net over F₈ — Constructive and digital

Digital (52, 108, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [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]

(52, 52+56, 160)-Net over F₈ — Digital

Digital (52, 108, 160)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+56, 4894)-Net in Base 8 — Upper bound on s

There is no (52, 108, 4895)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 279036 469964 123611 579910 516683 992865 121380 750557 689880 258489 367176 710234 368758 277854 566868 384803 > 8¹⁰⁸ [i]