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

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

(32, 32+37, 196)-Net over F₃₂ — Constructive and digital

Digital (32, 69, 196)-net over F₃₂, using

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

(32, 32+37, 288)-Net in Base 32 — Constructive

(32, 69, 288)-net in base 32, using

t-expansion [i] based on (31, 69, 288)-net in base 32, using
- 8 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
  - base change [i] based on digital (9, 55, 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]

(32, 32+37, 362)-Net over F₃₂ — Digital

Digital (32, 69, 362)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 32+37, 118265)-Net in Base 32 — Upper bound on s

There is no (32, 69, 118266)-net in base 32, because

1 times m-reduction [i] would yield (32, 68, 118266)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 239999 954647 139948 596602 330213 612380 310652 129605 644058 409917 669757 579724 882255 517758 340147 316118 625284 > 32⁶⁸ [i]