MinT - Best Known (21, 21+19, s)-Nets in Base 32

Best Known (21, 21+19, s)-Nets in Base 32

(21, 21+19, 175)-Net over F₃₂ — Constructive and digital

Digital (21, 40, 175)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 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, 10, 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, 26, 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]

(21, 21+19, 290)-Net in Base 32 — Constructive

(21, 40, 290)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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. (12, 31, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (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]

(21, 21+19, 649)-Net over F₃₂ — Digital

Digital (21, 40, 649)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁰, 649, F₃₂, 19) (dual of [649, 609, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴⁰, 1036, F₃₂, 19) (dual of [1036, 996, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(32³⁷, 1025, F₃₂, 19) (dual of [1025, 988, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(32²⁹, 1025, F₃₂, 15) (dual of [1025, 996, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (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]

(21, 21+19, 445350)-Net in Base 32 — Upper bound on s

There is no (21, 40, 445351)-net in base 32, because

1 times m-reduction [i] would yield (21, 39, 445351)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 50217 114240 582639 645293 266354 153472 351641 966785 202939 853352 > 32³⁹ [i]