MinT - Best Known (82−45, 82, s)-Nets in Base 32

Best Known (82−45, 82, s)-Nets in Base 32

(82−45, 82, 196)-Net over F₃₂ — Constructive and digital

Digital (37, 82, 196)-net over F₃₂, using

1 times m-reduction [i] based on digital (37, 83, 196)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (7, 53, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(82−45, 82, 288)-Net in Base 32 — Constructive

(37, 82, 288)-net in base 32, using

16 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(82−45, 82, 355)-Net over F₃₂ — Digital

Digital (37, 82, 355)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−45, 82, 101655)-Net in Base 32 — Upper bound on s

There is no (37, 82, 101656)-net in base 32, because

1 times m-reduction [i] would yield (37, 81, 101656)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 82 636238 268157 558271 186854 040612 157115 899747 891774 880821 126586 350741 765962 047009 585922 896374 460151 586815 456698 950528 534247 > 32⁸¹ [i]