MinT - Best Known (39, 66, s)-Nets in Base 32

Best Known (39, 66, s)-Nets in Base 32

(39, 66, 260)-Net over F₃₂ — Constructive and digital

Digital (39, 66, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 12, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. 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]
3. digital (7, 34, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(39, 66, 514)-Net in Base 32 — Constructive

(39, 66, 514)-net in base 32, using

32² times duplication [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)

(39, 66, 2266)-Net over F₃₂ — Digital

Digital (39, 66, 2266)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 66, 6134708)-Net in Base 32 — Upper bound on s

There is no (39, 66, 6134709)-net in base 32, because

1 times m-reduction [i] would yield (39, 65, 6134709)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 68 351677 799587 120891 422690 485989 246252 309040 388821 196208 076962 355731 083032 073414 222698 424171 074064 > 32⁶⁵ [i]