MinT - Best Known (33−15, 33, s)-Nets in Base 32

Best Known (33−15, 33, s)-Nets in Base 32

(33−15, 33, 175)-Net over F₃₂ — Constructive and digital

Digital (18, 33, 175)-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 (7, 22, 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]

(33−15, 33, 322)-Net in Base 32 — Constructive

(18, 33, 322)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (2, 9, 65)-net in base 32, using
  - 3 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
    - base change [i] based on digital (0, 10, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(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]

(33−15, 33, 921)-Net over F₃₂ — Digital

Digital (18, 33, 921)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³³, 921, F₃₂, 15) (dual of [921, 888, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(32³³, 1038, F₃₂, 15) (dual of [1038, 1005, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(9) [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(32¹⁹, 1024, F₃₂, 10) (dual of [1024, 1005, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [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]

(33−15, 33, 828401)-Net in Base 32 — Upper bound on s

There is no (18, 33, 828402)-net in base 32, because

1 times m-reduction [i] would yield (18, 32, 828402)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 461507 773148 847639 145862 123705 489009 860969 372216 > 32³² [i]