MinT - Best Known (57, 57+45, s)-Nets in Base 32

Best Known (57, 57+45, s)-Nets in Base 32

(57, 57+45, 272)-Net over F₃₂ — Constructive and digital

Digital (57, 102, 272)-net over F₃₂, using

2 times m-reduction [i] based on digital (57, 104, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 20, 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, 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)

(57, 57+45, 513)-Net in Base 32 — Constructive

(57, 102, 513)-net in base 32, using

t-expansion [i] based on (46, 102, 513)-net in base 32, using
- 6 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]

(57, 57+45, 1739)-Net over F₃₂ — Digital

Digital (57, 102, 1739)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+45, 2374083)-Net in Base 32 — Upper bound on s

There is no (57, 102, 2374084)-net in base 32, because

1 times m-reduction [i] would yield (57, 101, 2374084)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 104 748923 829399 232655 090493 256569 065251 371808 867174 404151 477675 553350 730101 814461 411364 217937 240692 344517 418022 661968 060494 980147 801289 994813 702184 832912 > 32¹⁰¹ [i]