MinT - Best Known (106−69, 106, s)-Nets in Base 32

Best Known (106−69, 106, s)-Nets in Base 32

(106−69, 106, 120)-Net over F₃₂ — Constructive and digital

Digital (37, 106, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 106, 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]

(106−69, 106, 216)-Net in Base 32 — Constructive

(37, 106, 216)-net in base 32, using

t-expansion [i] based on (36, 106, 216)-net in base 32, using
- 2 times m-reduction [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]

(106−69, 106, 273)-Net over F₃₂ — Digital

Digital (37, 106, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 106, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(106−69, 106, 315)-Net in Base 32

(37, 106, 315)-net in base 32, using

2 times m-reduction [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]

(106−69, 106, 19407)-Net in Base 32 — Upper bound on s

There is no (37, 106, 19408)-net in base 32, because

1 times m-reduction [i] would yield (37, 105, 19408)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 864893 917628 655945 418132 919003 404939 861637 422895 522742 600624 821458 358564 462254 362264 982356 845002 382938 337704 181466 591595 944553 168195 101455 760956 895595 577666 > 32¹⁰⁵ [i]