MinT - Best Known (32, 52, s)-Nets in Base 8

Best Known (32, 52, s)-Nets in Base 8

(32, 52, 256)-Net over F₈ — Constructive and digital

Digital (32, 52, 256)-net over F₈, using

2 times m-reduction [i] based on digital (32, 54, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 27, 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]

(32, 52, 300)-Net in Base 8 — Constructive

(32, 52, 300)-net in base 8, using

2 times m-reduction [i] based on (32, 54, 300)-net in base 8, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
  - 1 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]

(32, 52, 381)-Net over F₈ — Digital

Digital (32, 52, 381)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵², 381, F₈, 20) (dual of [381, 329, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵², 511, F₈, 20) (dual of [511, 459, 21]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(32, 52, 32126)-Net in Base 8 — Upper bound on s

There is no (32, 52, 32127)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 91352 493348 876861 523207 824764 481673 428967 715659 > 8⁵² [i]