MinT - Best Known (71−42, 71, s)-Nets in Base 32

Best Known (71−42, 71, s)-Nets in Base 32

(71−42, 71, 142)-Net over F₃₂ — Constructive and digital

Digital (29, 71, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 49, 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]

(71−42, 71, 257)-Net over F₃₂ — Digital

Digital (29, 71, 257)-net over F₃₂, using

t-expansion [i] based on digital (28, 71, 257)-net over F₃₂, using
- net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
    - a curve from the manYPoints database [i]

(71−42, 71, 259)-Net in Base 32 — Constructive

(29, 71, 259)-net in base 32, using

1 times m-reduction [i] based on (29, 72, 259)-net in base 32, using
- base change [i] based on digital (2, 45, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(71−42, 71, 321)-Net in Base 32

(29, 71, 321)-net in base 32, using

1 times m-reduction [i] based on (29, 72, 321)-net in base 32, using
- base change [i] based on digital (2, 45, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(71−42, 71, 34342)-Net in Base 32 — Upper bound on s

There is no (29, 71, 34343)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 73398 868135 760940 890772 086591 140077 111115 167117 448114 479525 985879 883728 187183 549118 985837 538200 235085 824428 > 32⁷¹ [i]