MinT - Best Known (83−35, 83, s)-Nets in Base 32

Best Known (83−35, 83, s)-Nets in Base 32

(83−35, 83, 273)-Net over F₃₂ — Constructive and digital

Digital (48, 83, 273)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 17, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 12, 44)-net over F₃₂, using
      - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
2. digital (7, 24, 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, 42, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(83−35, 83, 513)-Net in Base 32 — Constructive

(48, 83, 513)-net in base 32, using

t-expansion [i] based on (46, 83, 513)-net in base 32, using
- 25 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(83−35, 83, 2080)-Net over F₃₂ — Digital

Digital (48, 83, 2080)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−35, 83, 4214225)-Net in Base 32 — Upper bound on s

There is no (48, 83, 4214226)-net in base 32, because

1 times m-reduction [i] would yield (48, 82, 4214226)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2644 223942 755358 766859 616900 629743 064251 014717 051704 755616 466191 087857 199103 845786 858849 217891 982733 907592 322147 544194 331787 > 32⁸² [i]