MinT - Best Known (60−36, 60, s)-Nets in Base 32

Best Known (60−36, 60, s)-Nets in Base 32

(60−36, 60, 128)-Net over F₃₂ — Constructive and digital

Digital (24, 60, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 39, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(60−36, 60, 225)-Net over F₃₂ — Digital

Digital (24, 60, 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]

(60−36, 60, 258)-Net in Base 32 — Constructive

(24, 60, 258)-net in base 32, using

base change [i] based on (14, 50, 258)-net in base 64, using
- 2 times m-reduction [i] based on (14, 52, 258)-net in base 64, using
  - base change [i] based on digital (1, 39, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(60−36, 60, 289)-Net in Base 32

(24, 60, 289)-net in base 32, using

base change [i] based on (14, 50, 289)-net in base 64, using
- 2 times m-reduction [i] based on (14, 52, 289)-net in base 64, using
  - base change [i] based on digital (1, 39, 289)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
        sharp bound on the number of rational points on curves with genus 1 [i]

(60−36, 60, 25338)-Net in Base 32 — Upper bound on s

There is no (24, 60, 25339)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 037889 755249 430429 477890 610211 199374 670703 250998 179401 083667 613478 820505 691852 057119 399329 > 32⁶⁰ [i]