MinT - Best Known (50, 50+40, s)-Nets in Base 32

Best Known (50, 50+40, s)-Nets in Base 32

(50, 50+40, 260)-Net over F₃₂ — Constructive and digital

Digital (50, 90, 260)-net over F₃₂, using

1 times m-reduction [i] based on digital (50, 91, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 16, 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, 27, 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, 48, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(50, 50+40, 513)-Net in Base 32 — Constructive

(50, 90, 513)-net in base 32, using

t-expansion [i] based on (46, 90, 513)-net in base 32, using
- 18 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+40, 1497)-Net over F₃₂ — Digital

Digital (50, 90, 1497)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+40, 1588973)-Net in Base 32 — Upper bound on s

There is no (50, 90, 1588974)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 371552 688450 515340 448757 827966 730034 898052 418593 745711 674985 211298 282029 486640 788341 356518 653745 788942 873916 641090 297754 095716 615986 > 32⁹⁰ [i]