MinT - Best Known (31, 77, s)-Nets in Base 32

Best Known (31, 77, s)-Nets in Base 32

(31, 77, 142)-Net over F₃₂ — Constructive and digital

Digital (31, 77, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 24, 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, 53, 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]

(31, 77, 273)-Net over F₃₂ — Digital

Digital (31, 77, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 77, 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]

(31, 77, 288)-Net in Base 32 — Constructive

(31, 77, 288)-net in base 32, using

base change [i] based on digital (9, 55, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(31, 77, 321)-Net in Base 32

(31, 77, 321)-net in base 32, using

base change [i] based on (9, 55, 321)-net in base 128, using
- 1 times m-reduction [i] based on (9, 56, 321)-net in base 128, using
  - base change [i] based on digital (2, 49, 321)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(31, 77, 33259)-Net in Base 32 — Upper bound on s

There is no (31, 77, 33260)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 806429 112854 814389 252634 356020 468421 546881 349514 868106 030376 581840 870484 259108 461181 549565 423047 222654 267120 208268 > 32⁷⁷ [i]