MinT - Best Known (36−21, 36, s)-Nets in Base 32

Best Known (36−21, 36, s)-Nets in Base 32

(36−21, 36, 120)-Net over F₃₂ — Constructive and digital

Digital (15, 36, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 36, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(36−21, 36, 158)-Net over F₃₂ — Digital

Digital (15, 36, 158)-net over F₃₂, using

net from sequence [i] based on digital (15, 157)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 15 and N(F) ≥ 158, using
  - an algebraic function field by Beelen and Pellikaan [i]

(36−21, 36, 258)-Net in Base 32 — Constructive

(15, 36, 258)-net in base 32, using

base change [i] based on (9, 30, 258)-net in base 64, using
- 2 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
  - base change [i] based on digital (1, 24, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(36−21, 36, 289)-Net in Base 32

(15, 36, 289)-net in base 32, using

base change [i] based on (9, 30, 289)-net in base 64, using
- 2 times m-reduction [i] based on (9, 32, 289)-net in base 64, using
  - base change [i] based on digital (1, 24, 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]

(36−21, 36, 27074)-Net in Base 32 — Upper bound on s

There is no (15, 36, 27075)-net in base 32, because

1 times m-reduction [i] would yield (15, 35, 27075)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 47897 408741 352997 176137 694347 404378 658345 560200 860776 > 32³⁵ [i]