MinT - Best Known (72−45, 72, s)-Nets in Base 32

Best Known (72−45, 72, s)-Nets in Base 32

(72−45, 72, 120)-Net over F₃₂ — Constructive and digital

Digital (27, 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]

(72−45, 72, 225)-Net over F₃₂ — Digital

Digital (27, 72, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 72, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(72−45, 72, 257)-Net in Base 32 — Constructive

(27, 72, 257)-net in base 32, using

base change [i] based on digital (0, 45, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(72−45, 72, 258)-Net in Base 32

(27, 72, 258)-net in base 32, using

base change [i] based on digital (15, 60, 258)-net over F₆₄, using
- net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
    - a curve from the manYPoints database [i]

(72−45, 72, 21027)-Net in Base 32 — Upper bound on s

There is no (27, 72, 21028)-net in base 32, because

1 times m-reduction [i] would yield (27, 71, 21028)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 73423 883601 034661 168259 965871 761280 042650 622104 439018 758240 880655 187697 514786 122384 155606 810847 232889 436168 > 32⁷¹ [i]