MinT - Best Known (68, 68+86, s)-Nets in Base 8

Best Known (68, 68+86, s)-Nets in Base 8

(68, 68+86, 113)-Net over F₈ — Constructive and digital

Digital (68, 154, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 54, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 100, 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]

(68, 68+86, 160)-Net over F₈ — Digital

Digital (68, 154, 160)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+86, 161)-Net in Base 8

(68, 154, 161)-net in base 8, using

2 times m-reduction [i] based on (68, 156, 161)-net in base 8, using
- base change [i] based on digital (29, 117, 161)-net over F₁₆, using
  - net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(68, 68+86, 4109)-Net in Base 8 — Upper bound on s

There is no (68, 154, 4110)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11 910969 439883 629359 982696 300703 984627 768408 089756 109614 712544 303147 134083 493953 315943 877095 887926 158492 347861 550415 804032 036225 602378 437376 > 8¹⁵⁴ [i]