MinT - Best Known (86−37, 86, s)-Nets in Base 32

Best Known (86−37, 86, s)-Nets in Base 32

(86−37, 86, 272)-Net over F₃₂ — Constructive and digital

Digital (49, 86, 272)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 17, 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, 25, 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, 44, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(86−37, 86, 513)-Net in Base 32 — Constructive

(49, 86, 513)-net in base 32, using

t-expansion [i] based on (46, 86, 513)-net in base 32, using
- 22 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]

(86−37, 86, 1834)-Net over F₃₂ — Digital

Digital (49, 86, 1834)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−37, 86, 3121905)-Net in Base 32 — Upper bound on s

There is no (49, 86, 3121906)-net in base 32, because

1 times m-reduction [i] would yield (49, 85, 3121906)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 86 646098 075906 129238 076776 749847 384582 383527 320597 818149 324619 902664 781827 903665 353975 044105 950110 903252 675423 854518 934867 664400 > 32⁸⁵ [i]