MinT - Best Known (39, 39+65, s)-Nets in Base 32

Best Known (39, 39+65, s)-Nets in Base 32

(39, 39+65, 131)-Net over F₃₂ — Constructive and digital

Digital (39, 104, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 32, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 72, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(39, 39+65, 288)-Net in Base 32 — Constructive

(39, 104, 288)-net in base 32, using

1 times m-reduction [i] based on (39, 105, 288)-net in base 32, using
- base change [i] based on digital (9, 75, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(39, 39+65, 291)-Net over F₃₂ — Digital

Digital (39, 104, 291)-net over F₃₂, using

t-expansion [i] based on digital (38, 104, 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]

(39, 39+65, 342)-Net in Base 32

(39, 104, 342)-net in base 32, using

t-expansion [i] based on (38, 104, 342)-net in base 32, using
- 4 times m-reduction [i] based on (38, 108, 342)-net in base 32, using
  - base change [i] based on digital (20, 90, 342)-net over F₆₄, using
    - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
        a curve from the manYPoints database [i]

(39, 39+65, 28838)-Net in Base 32 — Upper bound on s

There is no (39, 104, 28839)-net in base 32, because

1 times m-reduction [i] would yield (39, 103, 28839)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107332 769144 365258 205866 142887 719840 458116 842855 018364 442751 632446 469358 485932 872173 260651 038134 026617 261081 391497 234932 938073 087597 985895 679960 456121 731125 > 32¹⁰³ [i]