MinT - Best Known (66−40, 66, s)-Nets in Base 32

Best Known (66−40, 66, s)-Nets in Base 32

(66−40, 66, 128)-Net over F₃₂ — Constructive and digital

Digital (26, 66, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 43, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(66−40, 66, 225)-Net over F₃₂ — Digital

Digital (26, 66, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 66, 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]

(66−40, 66, 258)-Net in Base 32 — Constructive

(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]

(66−40, 66, 289)-Net in Base 32

(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]

(66−40, 66, 24817)-Net in Base 32 — Upper bound on s

There is no (26, 66, 24818)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2187 733686 804162 347929 920984 365428 595430 283089 850596 212486 936165 441997 015313 486321 941561 172490 822552 > 32⁶⁶ [i]