MinT - Best Known (59, 59+41, s)-Nets in Base 32

Best Known (59, 59+41, s)-Nets in Base 32

(59, 59+41, 316)-Net over F₃₂ — Constructive and digital

Digital (59, 100, 316)-net over F₃₂, using

32¹ times duplication [i] based on digital (58, 99, 316)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 20, 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, 27, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (11, 52, 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]

(59, 59+41, 514)-Net in Base 32 — Constructive

(59, 100, 514)-net in base 32, using

32² times duplication [i] based on (57, 98, 514)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (12, 32, 257)-net in base 32, using
    - base change [i] based on digital (0, 20, 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. (25, 66, 257)-net in base 32, using
    - base change [i] based on (14, 55, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
        base change [i] based on digital (0, 42, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(59, 59+41, 2966)-Net over F₃₂ — Digital

Digital (59, 100, 2966)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+41, 7558512)-Net in Base 32 — Upper bound on s

There is no (59, 100, 7558513)-net in base 32, because

1 times m-reduction [i] would yield (59, 99, 7558513)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 102293 539337 042783 006749 363296 201685 846912 436406 973722 964189 518891 762242 952621 813165 451183 763039 673958 892167 464391 663364 813954 994991 817366 948668 159384 > 32⁹⁹ [i]