MinT - Best Known (57, 57+42, s)-Nets in Base 32

Best Known (57, 57+42, s)-Nets in Base 32

(57, 57+42, 294)-Net over F₃₂ — Constructive and digital

Digital (57, 99, 294)-net over F₃₂, using

2 times m-reduction [i] based on digital (57, 101, 294)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 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, 29, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (7, 51, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(57, 57+42, 513)-Net in Base 32 — Constructive

(57, 99, 513)-net in base 32, using

t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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]

(57, 57+42, 2264)-Net over F₃₂ — Digital

Digital (57, 99, 2264)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+42, 3490089)-Net in Base 32 — Upper bound on s

There is no (57, 99, 3490090)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 102293 659331 351763 854295 208411 579540 528742 602681 044410 453488 723925 939487 692782 207811 209851 255066 616054 344154 362056 803458 400187 345059 902090 563707 258844 > 32⁹⁹ [i]