MinT - Best Known (43−20, 43, s)-Nets in Base 32

Best Known (43−20, 43, s)-Nets in Base 32

(43−20, 43, 175)-Net over F₃₂ — Constructive and digital

Digital (23, 43, 175)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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, 11, 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]
2. digital (7, 27, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(43−20, 43, 301)-Net in Base 32 — Constructive

(23, 43, 301)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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]
2. (12, 32, 257)-net in base 32, using
  - base change [i] based on digital (0, 20, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(43−20, 43, 784)-Net over F₃₂ — Digital

Digital (23, 43, 784)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴³, 784, F₃₂, 20) (dual of [784, 741, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴³, 1038, F₃₂, 20) (dual of [1038, 995, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(14) [i] based on
    1. linear OA(32³⁹, 1024, F₃₂, 20) (dual of [1024, 985, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(32²⁹, 1024, F₃₂, 15) (dual of [1024, 995, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(32⁴, 14, F₃₂, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁴, 32, F₃₂, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
        Reedâ€“Solomon code RS(28,32) [i]

(43−20, 43, 433266)-Net in Base 32 — Upper bound on s

There is no (23, 43, 433267)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 52657 273164 982220 527892 580573 792107 247543 443062 902535 822181 098034 > 32⁴³ [i]