MinT - Best Known (59, 59+45, s)-Nets in Base 32

Best Known (59, 59+45, s)-Nets in Base 32

(59, 59+45, 294)-Net over F₃₂ — Constructive and digital

Digital (59, 104, 294)-net over F₃₂, using

2 times m-reduction [i] based on digital (59, 106, 294)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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]
  2. digital (7, 30, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (7, 54, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(59, 59+45, 513)-Net in Base 32 — Constructive

(59, 104, 513)-net in base 32, using

t-expansion [i] based on (46, 104, 513)-net in base 32, using
- 4 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]

(59, 59+45, 2032)-Net over F₃₂ — Digital

Digital (59, 104, 2032)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+45, 3253331)-Net in Base 32 — Upper bound on s

There is no (59, 104, 3253332)-net in base 32, because

1 times m-reduction [i] would yield (59, 103, 3253332)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107262 536262 803012 898862 326167 421981 734779 619204 549240 508646 144069 306440 620034 421018 555225 628947 665956 630058 771803 394789 774724 807262 390357 300511 282199 925744 > 32¹⁰³ [i]