MinT - Best Known (19, 19+17, s)-Nets in Base 32

Best Known (19, 19+17, s)-Nets in Base 32

(19, 19+17, 164)-Net over F₃₂ — Constructive and digital

Digital (19, 36, 164)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 66)-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 (0, 8, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
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]

(19, 19+17, 290)-Net in Base 32 — Constructive

(19, 36, 290)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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. (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]

(19, 19+17, 667)-Net over F₃₂ — Digital

Digital (19, 36, 667)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁶, 667, F₃₂, 17) (dual of [667, 631, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁶, 1036, F₃₂, 17) (dual of [1036, 1000, 18]-code), using
  - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(32³³, 1025, F₃₂, 17) (dual of [1025, 992, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    2. linear OA(32²⁵, 1025, F₃₂, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    3. linear OA(32³, 11, F₃₂, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
      - discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]

(19, 19+17, 467042)-Net in Base 32 — Upper bound on s

There is no (19, 36, 467043)-net in base 32, because

1 times m-reduction [i] would yield (19, 35, 467043)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 47891 222347 854987 476669 828202 498530 081651 614752 243140 > 32³⁵ [i]