MinT - Best Known (59, 59+47, s)-Nets in Base 32

Best Known (59, 59+47, s)-Nets in Base 32

(59, 59+47, 294)-Net over F₃₂ — Constructive and digital

Digital (59, 106, 294)-net over F₃₂, using

generalized (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, 30, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 54, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(59, 59+47, 513)-Net in Base 32 — Constructive

(59, 106, 513)-net in base 32, using

t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 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]

(59, 59+47, 1730)-Net over F₃₂ — Digital

Digital (59, 106, 1730)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+47, 2261669)-Net in Base 32 — Upper bound on s

There is no (59, 106, 2261670)-net in base 32, because

1 times m-reduction [i] would yield (59, 105, 2261670)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 837046 820259 548377 265508 360029 836524 227420 764648 382962 608351 182194 752953 270587 502048 089635 570123 421726 187934 975467 527232 437260 242536 621083 732706 316838 338672 > 32¹⁰⁵ [i]