MinT - Best Known (24, 45, s)-Nets in Base 32

Best Known (24, 45, s)-Nets in Base 32

(24, 45, 196)-Net over F₃₂ — Constructive and digital

Digital (24, 45, 196)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 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]
2. digital (7, 28, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(24, 45, 290)-Net in Base 32 — Constructive

(24, 45, 290)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 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. (14, 35, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (14, 36, 257)-net in base 32, using
    - base change [i] based on (8, 30, 257)-net in base 64, using
      - 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
        base change [i] based on digital (0, 24, 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]

(24, 45, 774)-Net over F₃₂ — Digital

Digital (24, 45, 774)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁵, 774, F₃₂, 21) (dual of [774, 729, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴⁵, 1038, F₃₂, 21) (dual of [1038, 993, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
    1. linear OA(32⁴¹, 1024, F₃₂, 21) (dual of [1024, 983, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(32³¹, 1024, F₃₂, 16) (dual of [1024, 993, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [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]

(24, 45, 612732)-Net in Base 32 — Upper bound on s

There is no (24, 45, 612733)-net in base 32, because

1 times m-reduction [i] would yield (24, 44, 612733)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 685003 451304 167253 160373 939623 627887 506100 823511 110251 623466 028798 > 32⁴⁴ [i]