MinT - Best Known (54, 54+53, s)-Nets in Base 32

Best Known (54, 54+53, s)-Nets in Base 32

(54, 54+53, 240)-Net over F₃₂ — Constructive and digital

Digital (54, 107, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 107, 240)-net over F₃₂, using
- 2 times m-reduction [i] based on digital (51, 109, 240)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 40, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]
    2. digital (11, 69, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(54, 54+53, 513)-Net in Base 32 — Constructive

(54, 107, 513)-net in base 32, using

t-expansion [i] based on (46, 107, 513)-net in base 32, using
- 1 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+53, 837)-Net over F₃₂ — Digital

Digital (54, 107, 837)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+53, 465907)-Net in Base 32 — Upper bound on s

There is no (54, 107, 465908)-net in base 32, because

1 times m-reduction [i] would yield (54, 106, 465908)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3514 897302 393823 614747 079135 004435 860986 870068 984370 520402 219024 876561 764169 269595 660723 981756 447300 012245 815123 792439 382184 550646 792832 622020 937772 959453 924172 > 32¹⁰⁶ [i]