MinT - Best Known (15−11, 15, s)-Nets in Base 32

Best Known (15−11, 15, s)-Nets in Base 32

(15−11, 15, 64)-Net over F₃₂ — Constructive and digital

Digital (4, 15, 64)-net over F₃₂, using

t-expansion [i] based on digital (3, 15, 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]

(15−11, 15, 71)-Net over F₃₂ — Digital

Digital (4, 15, 71)-net over F₃₂, using

net from sequence [i] based on digital (4, 70)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 4 and N(F) ≥ 71, using
  - a curve from the manYPoints database [i]

(15−11, 15, 80)-Net in Base 32 — Constructive

(4, 15, 80)-net in base 32, using

3 times m-reduction [i] based on (4, 18, 80)-net in base 32, using
- base change [i] based on digital (1, 15, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(15−11, 15, 81)-Net in Base 32

(4, 15, 81)-net in base 32, using

3 times m-reduction [i] based on (4, 18, 81)-net in base 32, using
- base change [i] based on digital (1, 15, 81)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(15−11, 15, 1374)-Net in Base 32 — Upper bound on s

There is no (4, 15, 1375)-net in base 32, because

1 times m-reduction [i] would yield (4, 14, 1375)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1181 815636 569729 741526 > 32¹⁴ [i]