MinT - Best Known (4, 4+7, s)-Nets in Base 32

Best Known (4, 4+7, s)-Nets in Base 32

(4, 4+7, 77)-Net over F₃₂ — Constructive and digital

Digital (4, 11, 77)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 8, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]

(4, 4+7, 85)-Net over F₃₂ — Digital

Digital (4, 11, 85)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹¹, 85, F₃₂, 7) (dual of [85, 74, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹¹, 93, F₃₂, 7) (dual of [93, 82, 8]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(4, 4+7, 129)-Net in Base 32 — Constructive

(4, 11, 129)-net in base 32, using

3 times m-reduction [i] based on (4, 14, 129)-net in base 32, using
- base change [i] based on digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(4, 4+7, 6096)-Net in Base 32 — Upper bound on s

There is no (4, 11, 6097)-net in base 32, because

1 times m-reduction [i] would yield (4, 10, 6097)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1125 943900 014480 > 32¹⁰ [i]