MinT - Best Known (23, 37, s)-Nets in Base 8

Best Known (23, 37, s)-Nets in Base 8

(23, 37, 208)-Net over F₈ — Constructive and digital

Digital (23, 37, 208)-net over F₈, using

3 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]

(23, 37, 300)-Net in Base 8 — Constructive

(23, 37, 300)-net in base 8, using

1 times m-reduction [i] based on (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]

(23, 37, 381)-Net over F₈ — Digital

Digital (23, 37, 381)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³⁷, 381, F₈, 14) (dual of [381, 344, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8³⁷, 511, F₈, 14) (dual of [511, 474, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(23, 37, 28657)-Net in Base 8 — Upper bound on s

There is no (23, 37, 28658)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2596 576757 018975 099572 911368 673720 > 8³⁷ [i]