MinT - Best Known (107−55, 107, s)-Nets in Base 32

Best Known (107−55, 107, s)-Nets in Base 32

(107−55, 107, 240)-Net over F₃₂ — Constructive and digital

Digital (52, 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)

(107−55, 107, 513)-Net in Base 32 — Constructive

(52, 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]

(107−55, 107, 657)-Net over F₃₂ — Digital

Digital (52, 107, 657)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−55, 107, 285845)-Net in Base 32 — Upper bound on s

There is no (52, 107, 285846)-net in base 32, because

1 times m-reduction [i] would yield (52, 106, 285846)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3514 950041 873632 875429 205832 172795 523270 149146 047779 536190 384072 347901 382702 745579 576924 945305 876544 453994 840444 257420 599000 453339 417626 383571 221829 743614 390240 > 32¹⁰⁶ [i]