MinT - Best Known (81−49, 81, s)-Nets in Base 32

Best Known (81−49, 81, s)-Nets in Base 32

(81−49, 81, 142)-Net over F₃₂ — Constructive and digital

Digital (32, 81, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 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, 56, 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]

(81−49, 81, 258)-Net in Base 32 — Constructive

(32, 81, 258)-net in base 32, using

32¹ times duplication [i] based on (31, 80, 258)-net in base 32, using
- base change [i] based on digital (1, 50, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(81−49, 81, 273)-Net over F₃₂ — Digital

Digital (32, 81, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 81, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(81−49, 81, 289)-Net in Base 32

(32, 81, 289)-net in base 32, using

32¹ times duplication [i] based on (31, 80, 289)-net in base 32, using
- base change [i] based on digital (1, 50, 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]

(81−49, 81, 32885)-Net in Base 32 — Upper bound on s

There is no (32, 81, 32886)-net in base 32, because

1 times m-reduction [i] would yield (32, 80, 32886)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 583029 971267 544758 935248 996649 279159 027517 121671 325181 680828 341547 268340 771284 451707 882251 871065 925265 991806 348678 137724 > 32⁸⁰ [i]