MinT - Best Known (56, 56+47, s)-Nets in Base 32

Best Known (56, 56+47, s)-Nets in Base 32

(56, 56+47, 260)-Net over F₃₂ — Constructive and digital

Digital (56, 103, 260)-net over F₃₂, using

32¹ times duplication [i] based on digital (55, 102, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 18, 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, 30, 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, 54, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(56, 56+47, 513)-Net in Base 32 — Constructive

(56, 103, 513)-net in base 32, using

t-expansion [i] based on (46, 103, 513)-net in base 32, using
- 5 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]

(56, 56+47, 1385)-Net over F₃₂ — Digital

Digital (56, 103, 1385)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+47, 1439143)-Net in Base 32 — Upper bound on s

There is no (56, 103, 1439144)-net in base 32, because

1 times m-reduction [i] would yield (56, 102, 1439144)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3351 998451 760135 472873 730535 211702 484844 652464 677228 256726 066334 008033 404627 182051 695468 964079 104345 913701 593646 368720 335347 077870 355336 581533 693078 938746 > 32¹⁰² [i]