MinT - Best Known (109−71, 109, s)-Nets in Base 32

Best Known (109−71, 109, s)-Nets in Base 32

(109−71, 109, 120)-Net over F₃₂ — Constructive and digital

Digital (38, 109, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 109, 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]

(109−71, 109, 216)-Net in Base 32 — Constructive

(38, 109, 216)-net in base 32, using

32¹ times duplication [i] based on (37, 108, 216)-net in base 32, using
- t-expansion [i] based on (36, 108, 216)-net in base 32, using
  - base change [i] based on (18, 90, 216)-net in base 64, using
    - 1 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
        net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(109−71, 109, 291)-Net over F₃₂ — Digital

Digital (38, 109, 291)-net over F₃₂, using

net from sequence [i] based on digital (38, 290)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 38 and N(F) ≥ 291, using
  - a curve from the manYPoints database [i]

(109−71, 109, 315)-Net in Base 32

(38, 109, 315)-net in base 32, using

32¹ times duplication [i] based on (37, 108, 315)-net in base 32, using
- base change [i] based on digital (19, 90, 315)-net over F₆₄, using
  - net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
      - a curve from the manYPoints database [i]

(109−71, 109, 19768)-Net in Base 32 — Upper bound on s

There is no (38, 109, 19769)-net in base 32, because

1 times m-reduction [i] would yield (38, 108, 19769)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 604303 810477 051483 172085 771711 877907 189771 200891 827264 144132 505905 294405 954141 326938 564112 945603 037198 991985 741065 464071 741557 647954 750125 086264 984376 063624 815688 > 32¹⁰⁸ [i]