MinT - Best Known (40−21, 40, s)-Nets in Base 32

Best Known (40−21, 40, s)-Nets in Base 32

(40−21, 40, 142)-Net over F₃₂ — Constructive and digital

Digital (19, 40, 142)-net over F₃₂, using

1 times m-reduction [i] based on digital (19, 41, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 12, 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, 29, 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]

(40−21, 40, 261)-Net in Base 32 — Constructive

(19, 40, 261)-net in base 32, using

base change [i] based on digital (4, 25, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(40−21, 40, 306)-Net over F₃₂ — Digital

Digital (19, 40, 306)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁰, 306, F₃₂, 21) (dual of [306, 266, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁴⁰, 341, F₃₂, 21) (dual of [341, 301, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 341Â | 32²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(40−21, 40, 321)-Net in Base 32

(19, 40, 321)-net in base 32, using

t-expansion [i] based on (18, 40, 321)-net in base 32, using
- 2 times m-reduction [i] based on (18, 42, 321)-net in base 32, using
  - base change [i] based on (11, 35, 321)-net in base 64, using
    - 1 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
      - base change [i] based on digital (2, 27, 321)-net over F₂₅₆, using
        net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(40−21, 40, 108312)-Net in Base 32 — Upper bound on s

There is no (19, 40, 108313)-net in base 32, because

1 times m-reduction [i] would yield (19, 39, 108313)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 50217 271859 404759 943221 744125 139538 121834 344304 788772 668328 > 32³⁹ [i]