MinT - Best Known (52, 52+39, s)-Nets in Base 32

Best Known (52, 52+39, s)-Nets in Base 32

(52, 52+39, 272)-Net over F₃₂ — Constructive and digital

Digital (52, 91, 272)-net over F₃₂, using

2 times m-reduction [i] based on digital (52, 93, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 18, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, 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, 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, 48, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(52, 52+39, 513)-Net in Base 32 — Constructive

(52, 91, 513)-net in base 32, using

t-expansion [i] based on (46, 91, 513)-net in base 32, using
- 17 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]

(52, 52+39, 1968)-Net over F₃₂ — Digital

Digital (52, 91, 1968)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+39, 3447548)-Net in Base 32 — Upper bound on s

There is no (52, 91, 3447549)-net in base 32, because

1 times m-reduction [i] would yield (52, 90, 3447549)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 361750 944162 945225 701450 693339 654961 719594 843201 097534 366920 801252 948426 607832 958894 744423 885543 379322 069702 894580 010400 124316 185092 > 32⁹⁰ [i]