MinT - Best Known (68, 68+34, s)-Nets in Base 8

Best Known (68, 68+34, s)-Nets in Base 8

(68, 68+34, 378)-Net over F₈ — Constructive and digital

Digital (68, 102, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (48, 82, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 41, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(68, 68+34, 576)-Net in Base 8 — Constructive

(68, 102, 576)-net in base 8, using

trace code for nets [i] based on (17, 51, 288)-net in base 64, using
- 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(68, 68+34, 1180)-Net over F₈ — Digital

Digital (68, 102, 1180)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+34, 268763)-Net in Base 8 — Upper bound on s

There is no (68, 102, 268764)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 130 374682 553158 899484 464719 857234 808372 053569 048214 121703 999875 095131 365787 771385 561880 109007 > 8¹⁰² [i]