MinT - Best Known (72−51, 72, s)-Nets in Base 32

Best Known (72−51, 72, s)-Nets in Base 32

(72−51, 72, 120)-Net over F₃₂ — Constructive and digital

Digital (21, 72, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 72, 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]

(72−51, 72, 177)-Net in Base 32 — Constructive

(21, 72, 177)-net in base 32, using

12 times m-reduction [i] based on (21, 84, 177)-net in base 32, using
- base change [i] based on digital (7, 70, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(72−51, 72, 185)-Net over F₃₂ — Digital

Digital (21, 72, 185)-net over F₃₂, using

net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
  - a curve from the manYPoints database [i]

(72−51, 72, 209)-Net in Base 32

(21, 72, 209)-net in base 32, using

base change [i] based on digital (9, 60, 209)-net over F₆₄, using
- net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
    - a curve from the manYPoints database [i]

(72−51, 72, 6165)-Net in Base 32 — Upper bound on s

There is no (21, 72, 6166)-net in base 32, because

1 times m-reduction [i] would yield (21, 71, 6166)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 73451 593028 071337 321063 389065 157126 360539 556928 837839 642248 023803 087156 309741 264327 540694 401038 016521 180928 > 32⁷¹ [i]