MinT - Best Known (61, 61+43, s)-Nets in Base 32

Best Known (61, 61+43, s)-Nets in Base 32

(61, 61+43, 316)-Net over F₃₂ — Constructive and digital

Digital (61, 104, 316)-net over F₃₂, using

1 times m-reduction [i] based on digital (61, 105, 316)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 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]
  2. digital (7, 29, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (11, 55, 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]

(61, 61+43, 515)-Net in Base 32 — Constructive

(61, 104, 515)-net in base 32, using

base change [i] based on digital (22, 65, 515)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 21, 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 (1, 44, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(61, 61+43, 2862)-Net over F₃₂ — Digital

Digital (61, 104, 2862)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+43, 6753555)-Net in Base 32 — Upper bound on s

There is no (61, 104, 6753556)-net in base 32, because

1 times m-reduction [i] would yield (61, 103, 6753556)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107262 691511 988961 851395 451496 885719 554130 984069 414164 056317 987238 051809 046077 440009 523946 861774 824770 053339 002216 883653 917579 007644 199635 696432 507195 114816 > 32¹⁰³ [i]