MinT - Best Known (66−36, 66, s)-Nets in Base 32

Best Known (66−36, 66, s)-Nets in Base 32

(66−36, 66, 174)-Net over F₃₂ — Constructive and digital

Digital (30, 66, 174)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 43, 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]

(66−36, 66, 288)-Net in Base 32 — Constructive

(30, 66, 288)-net in base 32, using

6 times m-reduction [i] based on (30, 72, 288)-net in base 32, using
- base change [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(66−36, 66, 315)-Net over F₃₂ — Digital

Digital (30, 66, 315)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−36, 66, 321)-Net in Base 32

(30, 66, 321)-net in base 32, using

t-expansion [i] based on (29, 66, 321)-net in base 32, using
- 6 times m-reduction [i] based on (29, 72, 321)-net in base 32, using
  - base change [i] based on digital (2, 45, 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]

(66−36, 66, 80464)-Net in Base 32 — Upper bound on s

There is no (30, 66, 80465)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2187 650180 129846 362013 076176 156517 593738 264990 944440 355708 322631 439587 033924 710845 142470 851205 384288 > 32⁶⁶ [i]