MinT - Best Known (37, 37+26, s)-Nets in Base 32

Best Known (37, 37+26, s)-Nets in Base 32

(37, 37+26, 240)-Net over F₃₂ — Constructive and digital

Digital (37, 63, 240)-net over F₃₂, using

4 times m-reduction [i] based on digital (37, 67, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 26, 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]
  2. digital (11, 41, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(37, 37+26, 514)-Net in Base 32 — Constructive

(37, 63, 514)-net in base 32, using

1 times m-reduction [i] based on (37, 64, 514)-net in base 32, using
- base change [i] based on digital (13, 40, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 13, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 27, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(37, 37+26, 2051)-Net over F₃₂ — Digital

Digital (37, 63, 2051)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+26, 3599416)-Net in Base 32 — Upper bound on s

There is no (37, 63, 3599417)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 66749 789101 170907 597051 351737 857413 736608 658439 041352 969868 971798 905526 568912 915589 555554 488720 > 32⁶³ [i]