MinT - Best Known (55, 55+42, s)-Nets in Base 32

Best Known (55, 55+42, s)-Nets in Base 32

(55, 55+42, 272)-Net over F₃₂ — Constructive and digital

Digital (55, 97, 272)-net over F₃₂, using

2 times m-reduction [i] based on digital (55, 99, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 19, 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, 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]
  3. digital (7, 51, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(55, 55+42, 513)-Net in Base 32 — Constructive

(55, 97, 513)-net in base 32, using

t-expansion [i] based on (46, 97, 513)-net in base 32, using
- 11 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]

(55, 55+42, 1915)-Net over F₃₂ — Digital

Digital (55, 97, 1915)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+42, 2508929)-Net in Base 32 — Upper bound on s

There is no (55, 97, 2508930)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 99 896415 960171 141458 030158 696986 709357 168424 417736 403946 269871 266435 314901 119940 727317 616715 021781 176370 304135 136126 885258 038313 614736 014683 042336 > 32⁹⁷ [i]