MinT - Best Known (135−74, 135, s)-Nets in Base 8

Best Known (135−74, 135, s)-Nets in Base 8

(135−74, 135, 111)-Net over F₈ — Constructive and digital

Digital (61, 135, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 47, 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, 88, 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]

(135−74, 135, 154)-Net over F₈ — Digital

Digital (61, 135, 154)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−74, 135, 156)-Net in Base 8

(61, 135, 156)-net in base 8, using

1 times m-reduction [i] based on (61, 136, 156)-net in base 8, using
- base change [i] based on digital (27, 102, 156)-net over F₁₆, using
  - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
      - a curve from the manYPoints database [i]

(135−74, 135, 4106)-Net in Base 8 — Upper bound on s

There is no (61, 135, 4107)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 83 367808 370388 619379 868965 485217 042127 924947 323654 884311 527212 546402 433891 960190 651008 988182 333832 871039 969507 335795 795456 > 8¹³⁵ [i]