MinT - Best Known (45−17, 45, s)-Nets in Base 8

Best Known (45−17, 45, s)-Nets in Base 8

(45−17, 45, 256)-Net over F₈ — Constructive and digital

Digital (28, 45, 256)-net over F₈, using

1 times m-reduction [i] based on digital (28, 46, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 23, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(45−17, 45, 300)-Net in Base 8 — Constructive

(28, 45, 300)-net in base 8, using

1 times m-reduction [i] based on (28, 46, 300)-net in base 8, using
- trace code for nets [i] based on (5, 23, 150)-net in base 64, using
  - 5 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
    - base change [i] based on digital (1, 24, 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]

(45−17, 45, 401)-Net over F₈ — Digital

Digital (28, 45, 401)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁵, 401, F₈, 17) (dual of [401, 356, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴⁵, 511, F₈, 17) (dual of [511, 466, 18]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(45−17, 45, 49836)-Net in Base 8 — Upper bound on s

There is no (28, 45, 49837)-net in base 8, because

1 times m-reduction [i] would yield (28, 44, 49837)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5445 058848 248976 977889 134647 161520 415747 > 8⁴⁴ [i]