MinT - Best Known (40−16, 40, s)-Nets in Base 8

Best Known (40−16, 40, s)-Nets in Base 8

(40−16, 40, 208)-Net over F₈ — Constructive and digital

Digital (24, 40, 208)-net over F₈, using

2 times m-reduction [i] based on digital (24, 42, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 21, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(40−16, 40, 258)-Net over F₈ — Digital

Digital (24, 40, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 20, 129)-net over F₆₄, using
- net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(40−16, 40, 300)-Net in Base 8 — Constructive

(24, 40, 300)-net in base 8, using

trace code for nets [i] based on (4, 20, 150)-net in base 64, using
- 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
  - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(40−16, 40, 17616)-Net in Base 8 — Upper bound on s

There is no (24, 40, 17617)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 329336 337460 156987 567956 303521 057134 > 8⁴⁰ [i]