MinT - Best Known (54, 54+39, s)-Nets in Base 32

Best Known (54, 54+39, s)-Nets in Base 32

(54, 54+39, 294)-Net over F₃₂ — Constructive and digital

Digital (54, 93, 294)-net over F₃₂, using

2 times m-reduction [i] based on digital (54, 95, 294)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 20, 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, 27, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (7, 48, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(54, 54+39, 513)-Net in Base 32 — Constructive

(54, 93, 513)-net in base 32, using

t-expansion [i] based on (46, 93, 513)-net in base 32, using
- 15 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]

(54, 54+39, 2358)-Net over F₃₂ — Digital

Digital (54, 93, 2358)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+39, 4965324)-Net in Base 32 — Upper bound on s

There is no (54, 93, 4965325)-net in base 32, because

1 times m-reduction [i] would yield (54, 92, 4965325)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 977141 368850 313445 803907 303071 530452 060210 757738 200124 688534 986093 862254 086691 347889 836704 653772 298360 625383 247127 622244 128750 621516 293936 > 32⁹² [i]