MinT - Best Known (20, 20+17, s)-Nets in Base 32

Best Known (20, 20+17, s)-Nets in Base 32

(20, 20+17, 175)-Net over F₃₂ — Constructive and digital

Digital (20, 37, 175)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 13, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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, 9, 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, 24, 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]

(20, 20+17, 301)-Net in Base 32 — Constructive

(20, 37, 301)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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. (11, 28, 257)-net in base 32, using
  - base change [i] based on (3, 20, 257)-net in base 128, using
    - 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
      - base change [i] based on digital (0, 21, 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]

(20, 20+17, 842)-Net over F₃₂ — Digital

Digital (20, 37, 842)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁷, 842, F₃₂, 17) (dual of [842, 805, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁷, 1038, F₃₂, 17) (dual of [1038, 1001, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(11) [i] based on
    1. linear OA(32³³, 1024, F₃₂, 17) (dual of [1024, 991, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(32²³, 1024, F₃₂, 12) (dual of [1024, 1001, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [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]

(20, 20+17, 720279)-Net in Base 32 — Upper bound on s

There is no (20, 37, 720280)-net in base 32, because

1 times m-reduction [i] would yield (20, 36, 720280)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 532505 358814 264025 490574 743562 413038 232966 646585 004922 > 32³⁶ [i]