MinT - Best Known (68−20, 68, s)-Nets in Base 8

Best Known (68−20, 68, s)-Nets in Base 8

(68−20, 68, 379)-Net over F₈ — Constructive and digital

Digital (48, 68, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (34, 54, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 27, 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]

(68−20, 68, 576)-Net in Base 8 — Constructive

(48, 68, 576)-net in base 8, using

trace code for nets [i] based on (14, 34, 288)-net in base 64, using
- 1 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
  - base change [i] based on digital (9, 30, 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]

(68−20, 68, 1942)-Net over F₈ — Digital

Digital (48, 68, 1942)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−20, 68, 895132)-Net in Base 8 — Upper bound on s

There is no (48, 68, 895133)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 711168 367608 877422 704963 069031 573273 323702 243817 618632 641766 > 8⁶⁸ [i]