MinT - Best Known (28, 69, s)-Nets in Base 32

Best Known (28, 69, s)-Nets in Base 32

(28, 69, 142)-Net over F₃₂ — Constructive and digital

Digital (28, 69, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 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, 48, 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]

(28, 69, 257)-Net over F₃₂ — Digital

Digital (28, 69, 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]

(28, 69, 258)-Net in Base 32 — Constructive

(28, 69, 258)-net in base 32, using

3 times m-reduction [i] based on (28, 72, 258)-net in base 32, using
- base change [i] based on digital (1, 45, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(28, 69, 289)-Net in Base 32

(28, 69, 289)-net in base 32, using

3 times m-reduction [i] based on (28, 72, 289)-net in base 32, using
- base change [i] based on digital (1, 45, 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]

(28, 69, 35101)-Net in Base 32 — Upper bound on s

There is no (28, 69, 35102)-net in base 32, because

1 times m-reduction [i] would yield (28, 68, 35102)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 240043 047130 074297 090202 742751 139988 123601 383538 261433 985509 651506 245983 652818 451861 015055 608489 919501 > 32⁶⁸ [i]