MinT - Best Known (26, 26+37, s)-Nets in Base 32

Best Known (26, 26+37, s)-Nets in Base 32

(26, 26+37, 142)-Net over F₃₂ — Constructive and digital

Digital (26, 63, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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, 44, 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]

(26, 26+37, 225)-Net over F₃₂ — Digital

Digital (26, 63, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 63, 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]

(26, 26+37, 259)-Net in Base 32 — Constructive

(26, 63, 259)-net in base 32, using

1 times m-reduction [i] based on (26, 64, 259)-net in base 32, using
- base change [i] based on digital (2, 40, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(26, 26+37, 321)-Net in Base 32

(26, 63, 321)-net in base 32, using

1 times m-reduction [i] based on (26, 64, 321)-net in base 32, using
- base change [i] based on digital (2, 40, 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]

(26, 26+37, 37244)-Net in Base 32 — Upper bound on s

There is no (26, 63, 37245)-net in base 32, because

1 times m-reduction [i] would yield (26, 62, 37245)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2085 924911 570003 924718 827249 005746 121951 941658 701260 267328 290912 258455 308479 022924 794377 280738 > 32⁶² [i]