MinT - Best Known (56, 56+41, s)-Nets in Base 32

Best Known (56, 56+41, s)-Nets in Base 32

(56, 56+41, 300)-Net over F₃₂ — Constructive and digital

Digital (56, 97, 300)-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 (9, 50, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(56, 56+41, 513)-Net in Base 32 — Constructive

(56, 97, 513)-net in base 32, using

t-expansion [i] based on (46, 97, 513)-net in base 32, using
- 11 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]

(56, 56+41, 2292)-Net over F₃₂ — Digital

Digital (56, 97, 2292)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+41, 4494314)-Net in Base 32 — Upper bound on s

There is no (56, 97, 4494315)-net in base 32, because

1 times m-reduction [i] would yield (56, 96, 4494315)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 121756 546127 842695 274950 576578 203304 268097 063104 840359 517290 940846 720514 336057 554559 142187 227039 788998 982163 141438 547691 250153 517807 253905 490564 > 32⁹⁶ [i]