MinT - Best Known (50, 109, s)-Nets in Base 32

Best Known (50, 109, s)-Nets in Base 32

(50, 109, 224)-Net over F₃₂ — Constructive and digital

Digital (50, 109, 224)-net over F₃₂, using

1 times m-reduction [i] based on digital (50, 110, 224)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 39, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (11, 71, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(50, 109, 483)-Net over F₃₂ — Digital

Digital (50, 109, 483)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 109, 513)-Net in Base 32 — Constructive

(50, 109, 513)-net in base 32, using

32¹ times duplication [i] based on (49, 108, 513)-net in base 32, using
- t-expansion [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, 109, 151741)-Net in Base 32 — Upper bound on s

There is no (50, 109, 151742)-net in base 32, because

1 times m-reduction [i] would yield (50, 108, 151742)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 599632 519955 864594 685933 828762 276171 630941 385618 788885 821054 951285 952192 855073 357564 677432 015246 333503 479801 134224 976698 672857 200554 801500 897025 142076 839268 529413 > 32¹⁰⁸ [i]