MinT - Best Known (29, 29+20, s)-Nets in Base 32

Best Known (29, 29+20, s)-Nets in Base 32

(29, 29+20, 226)-Net over F₃₂ — Constructive and digital

Digital (29, 49, 226)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 9, 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 (3, 13, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)
3. digital (7, 27, 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]

(29, 29+20, 514)-Net in Base 32 — Constructive

(29, 49, 514)-net in base 32, using

32¹ times duplication [i] based on (28, 48, 514)-net in base 32, using
- base change [i] based on digital (10, 30, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 10, 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, 20, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(29, 29+20, 1957)-Net over F₃₂ — Digital

Digital (29, 49, 1957)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(29, 29+20, 3466162)-Net in Base 32 — Upper bound on s

There is no (29, 49, 3466163)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 56 539214 184397 508719 243202 178900 976212 676286 643521 159957 527564 116053 407338 > 32⁴⁹ [i]