MinT - Best Known (19, 44, s)-Nets in Base 32

Best Known (19, 44, s)-Nets in Base 32

(19, 44, 131)-Net over F₃₂ — Constructive and digital

Digital (19, 44, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 32, 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]

(19, 44, 189)-Net over F₃₂ — Digital

Digital (19, 44, 189)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(19, 44, 258)-Net in Base 32 — Constructive

(19, 44, 258)-net in base 32, using

4 times m-reduction [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]

(19, 44, 289)-Net in Base 32

(19, 44, 289)-net in base 32, using

4 times m-reduction [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]

(19, 44, 42207)-Net in Base 32 — Upper bound on s

There is no (19, 44, 42208)-net in base 32, because

1 times m-reduction [i] would yield (19, 43, 42208)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 52658 376975 049246 619587 282597 581569 042667 320172 795507 544359 136469 > 32⁴³ [i]