MinT - Best Known (60−31, 60, s)-Nets in Base 32

Best Known (60−31, 60, s)-Nets in Base 32

(60−31, 60, 196)-Net over F₃₂ — Constructive and digital

Digital (29, 60, 196)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 22, 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, 38, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(60−31, 60, 288)-Net in Base 32 — Constructive

(29, 60, 288)-net in base 32, using

10 times m-reduction [i] based on (29, 70, 288)-net in base 32, using
- base change [i] based on digital (9, 50, 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]

(60−31, 60, 413)-Net over F₃₂ — Digital

Digital (29, 60, 413)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(60−31, 60, 172441)-Net in Base 32 — Upper bound on s

There is no (29, 60, 172442)-net in base 32, because

1 times m-reduction [i] would yield (29, 59, 172442)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 63658 188659 329244 081435 664304 656884 038396 163154 240335 251830 008976 986610 422550 410928 229288 > 32⁵⁹ [i]