MinT - Best Known (56, 101, s)-Nets in Base 32

Best Known (56, 101, s)-Nets in Base 32

(56, 101, 272)-Net over F₃₂ — Constructive and digital

Digital (56, 101, 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, 29, 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, 52, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(56, 101, 513)-Net in Base 32 — Constructive

(56, 101, 513)-net in base 32, using

t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 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, 101, 1609)-Net over F₃₂ — Digital

Digital (56, 101, 1609)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 101, 2028055)-Net in Base 32 — Upper bound on s

There is no (56, 101, 2028056)-net in base 32, because

1 times m-reduction [i] would yield (56, 100, 2028056)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 273423 718248 137784 409048 492465 267078 508915 365410 735971 974625 366475 772304 540646 093456 146996 223180 797211 813348 067924 439482 013787 710495 874970 471622 262327 > 32¹⁰⁰ [i]