MinT - Best Known (50, 50+37, s)-Nets in Base 32

Best Known (50, 50+37, s)-Nets in Base 32

(50, 50+37, 272)-Net over F₃₂ — Constructive and digital

Digital (50, 87, 272)-net over F₃₂, using

1 times m-reduction [i] based on digital (50, 88, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 17, 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]
  2. digital (7, 26, 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]
  3. digital (7, 45, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(50, 50+37, 513)-Net in Base 32 — Constructive

(50, 87, 513)-net in base 32, using

t-expansion [i] based on (46, 87, 513)-net in base 32, using
- 21 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(50, 50+37, 2017)-Net over F₃₂ — Digital

Digital (50, 87, 2017)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+37, 3784769)-Net in Base 32 — Upper bound on s

There is no (50, 87, 3784770)-net in base 32, because

1 times m-reduction [i] would yield (50, 86, 3784770)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2772 675797 796134 619146 541215 816356 732671 027173 230921 364199 024832 195268 594202 133112 973040 346011 288790 828950 484725 080535 836624 544928 > 32⁸⁶ [i]