MinT - Best Known (65−39, 65, s)-Nets in Base 32

Best Known (65−39, 65, s)-Nets in Base 32

(65−39, 65, 131)-Net over F₃₂ — Constructive and digital

Digital (26, 65, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 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, 46, 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]

(65−39, 65, 225)-Net over F₃₂ — Digital

Digital (26, 65, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 65, 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]

(65−39, 65, 258)-Net in Base 32 — Constructive

(26, 65, 258)-net in base 32, using

1 times m-reduction [i] based on (26, 66, 258)-net in base 32, using
- base change [i] based on (15, 55, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 56, 258)-net in base 64, using
    - base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(65−39, 65, 289)-Net in Base 32

(26, 65, 289)-net in base 32, using

1 times m-reduction [i] based on (26, 66, 289)-net in base 32, using
- base change [i] based on (15, 55, 289)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 56, 289)-net in base 64, using
    - base change [i] based on digital (1, 42, 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]

(65−39, 65, 30039)-Net in Base 32 — Upper bound on s

There is no (26, 65, 30040)-net in base 32, because

1 times m-reduction [i] would yield (26, 64, 30040)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 136565 086126 307894 056731 448054 918087 601974 936522 910690 449893 704056 347294 785083 970601 261291 150533 > 32⁶⁴ [i]