MinT - Best Known (38−15, 38, s)-Nets in Base 8

Best Known (38−15, 38, s)-Nets in Base 8

(38−15, 38, 208)-Net over F₈ — Constructive and digital

Digital (23, 38, 208)-net over F₈, using

2 times m-reduction [i] based on digital (23, 40, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 20, 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]

(38−15, 38, 258)-Net over F₈ — Digital

Digital (23, 38, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 19, 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]

(38−15, 38, 300)-Net in Base 8 — Constructive

(23, 38, 300)-net in base 8, using

trace code for nets [i] based on (4, 19, 150)-net in base 64, using
- 2 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]

(38−15, 38, 28657)-Net in Base 8 — Upper bound on s

There is no (23, 38, 28658)-net in base 8, because

1 times m-reduction [i] would yield (23, 37, 28658)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2596 576757 018975 099572 911368 673720 > 8³⁷ [i]