MinT - Best Known (49−29, 49, s)-Nets in Base 32

Best Known (49−29, 49, s)-Nets in Base 32

(49−29, 49, 128)-Net over F₃₂ — Constructive and digital

Digital (20, 49, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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, 32, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(49−29, 49, 177)-Net over F₃₂ — Digital

Digital (20, 49, 177)-net over F₃₂, using

net from sequence [i] based on digital (20, 176)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 20 and N(F) ≥ 177, using
  - a curve from the manYPoints database [i]

(49−29, 49, 258)-Net in Base 32 — Constructive

(20, 49, 258)-net in base 32, using

32¹ times duplication [i] based on (19, 48, 258)-net in base 32, using
- base change [i] based on digital (1, 30, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(49−29, 49, 289)-Net in Base 32

(20, 49, 289)-net in base 32, using

32¹ times duplication [i] based on (19, 48, 289)-net in base 32, using
- base change [i] based on digital (1, 30, 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]

(49−29, 49, 28216)-Net in Base 32 — Upper bound on s

There is no (20, 49, 28217)-net in base 32, because

1 times m-reduction [i] would yield (20, 48, 28217)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 767277 470532 793738 438486 861870 933375 658213 812212 289271 984007 196438 507672 > 32⁴⁸ [i]