MinT - Best Known (21, 21+18, s)-Nets in Base 32

Best Known (21, 21+18, s)-Nets in Base 32

(21, 21+18, 175)-Net over F₃₂ — Constructive and digital

Digital (21, 39, 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, 25, 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+18, 290)-Net in Base 32 — Constructive

(21, 39, 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, 30, 257)-net in base 32, using
  - 2 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+18, 817)-Net over F₃₂ — Digital

Digital (21, 39, 817)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁹, 817, F₃₂, 18) (dual of [817, 778, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(32³⁹, 1038, F₃₂, 18) (dual of [1038, 999, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(12) [i] based on
    1. linear OA(32³⁵, 1024, F₃₂, 18) (dual of [1024, 989, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(32²⁵, 1024, F₃₂, 13) (dual of [1024, 999, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(21, 21+18, 445350)-Net in Base 32 — Upper bound on s

There is no (21, 39, 445351)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 50217 114240 582639 645293 266354 153472 351641 966785 202939 853352 > 32³⁹ [i]