MinT - Best Known (16, 31, s)-Nets in Base 32

Best Known (16, 31, s)-Nets in Base 32

(16, 31, 152)-Net over F₃₂ — Constructive and digital

Digital (16, 31, 152)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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, 8, 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 (5, 20, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(16, 31, 290)-Net in Base 32 — Constructive

(16, 31, 290)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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. (9, 24, 257)-net in base 32, using
  - base change [i] based on digital (0, 15, 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]

(16, 31, 538)-Net over F₃₂ — Digital

Digital (16, 31, 538)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³¹, 538, F₃₂, 15) (dual of [538, 507, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(32³¹, 1032, F₃₂, 15) (dual of [1032, 1001, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(11) [i] based on
    1. 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]
    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(32², 8, F₃₂, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
      - discarding factors / shortening the dual code based on linear OA(32², 32, F₃₂, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
        Reedâ€“Solomon code RS(30,32) [i]

(16, 31, 307747)-Net in Base 32 — Upper bound on s

There is no (16, 31, 307748)-net in base 32, because

1 times m-reduction [i] would yield (16, 30, 307748)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1427 249371 549302 876235 388662 631652 805998 258648 > 32³⁰ [i]