MinT - Best Known (24, 51, s)-Nets in Base 32

Best Known (24, 51, s)-Nets in Base 32

(24, 51, 162)-Net over F₃₂ — Constructive and digital

Digital (24, 51, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (24, 52, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 17, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (7, 35, 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]

(24, 51, 260)-Net in Base 32 — Constructive

(24, 51, 260)-net in base 32, using

5 times m-reduction [i] based on (24, 56, 260)-net in base 32, using
- base change [i] based on digital (3, 35, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(24, 51, 326)-Net over F₃₂ — Digital

Digital (24, 51, 326)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁵¹, 326, F₃₂, 27) (dual of [326, 275, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁵¹, 341, F₃₂, 27) (dual of [341, 290, 28]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(24, 51, 112475)-Net in Base 32 — Upper bound on s

There is no (24, 51, 112476)-net in base 32, because

1 times m-reduction [i] would yield (24, 50, 112476)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1809 348125 848975 972882 957799 598594 878247 998250 615085 386997 870134 101938 109492 > 32⁵⁰ [i]