MinT - Best Known (51, 51+41, s)-Nets in Base 32

Best Known (51, 51+41, s)-Nets in Base 32

(51, 51+41, 260)-Net over F₃₂ — Constructive and digital

Digital (51, 92, 260)-net over F₃₂, using

32¹ times duplication [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)

(51, 51+41, 513)-Net in Base 32 — Constructive

(51, 92, 513)-net in base 32, using

t-expansion [i] based on (46, 92, 513)-net in base 32, using
- 16 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]

(51, 51+41, 1493)-Net over F₃₂ — Digital

Digital (51, 92, 1493)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+41, 1889620)-Net in Base 32 — Upper bound on s

There is no (51, 92, 1889621)-net in base 32, because

1 times m-reduction [i] would yield (51, 91, 1889621)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 93035 785420 257780 779981 214399 393605 515927 099084 947794 177423 621058 229619 080536 803763 153434 796759 290664 804247 302296 373850 106892 147787 417421 > 32⁹¹ [i]