MinT - Best Known (39, 68, s)-Nets in Base 32

Best Known (39, 68, s)-Nets in Base 32

(39, 68, 240)-Net over F₃₂ — Constructive and digital

Digital (39, 68, 240)-net over F₃₂, using

5 times m-reduction [i] based on digital (39, 73, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 28, 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, 45, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(39, 68, 407)-Net in Base 32 — Constructive

(39, 68, 407)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (7, 21, 150)-net in base 32, using
  - base change [i] based on digital (1, 15, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. (18, 47, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
    - base change [i] based on digital (0, 30, 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]

(39, 68, 1662)-Net over F₃₂ — Digital

Digital (39, 68, 1662)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 68, 3113958)-Net in Base 32 — Upper bound on s

There is no (39, 68, 3113959)-net in base 32, because

1 times m-reduction [i] would yield (39, 67, 3113959)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 69992 160047 440813 629497 153492 380245 400652 126466 537282 679676 364494 303265 962517 500917 551641 197831 053516 > 32⁶⁷ [i]