MinT - Best Known (90−39, 90, s)-Nets in Base 32

Best Known (90−39, 90, s)-Nets in Base 32

(90−39, 90, 272)-Net over F₃₂ — Constructive and digital

Digital (51, 90, 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, 26, 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, 46, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(90−39, 90, 513)-Net in Base 32 — Constructive

(51, 90, 513)-net in base 32, using

t-expansion [i] based on (46, 90, 513)-net in base 32, using
- 18 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]

(90−39, 90, 1798)-Net over F₃₂ — Digital

Digital (51, 90, 1798)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−39, 90, 2872709)-Net in Base 32 — Upper bound on s

There is no (51, 90, 2872710)-net in base 32, because

1 times m-reduction [i] would yield (51, 89, 2872710)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 855058 284427 008241 923755 761424 907820 643915 732206 730745 907663 821257 510101 942258 052718 195403 934101 039200 741387 977731 304576 094717 330800 > 32⁸⁹ [i]