MinT - Best Known (57, 57+48, s)-Nets in Base 32

Best Known (57, 57+48, s)-Nets in Base 32

(57, 57+48, 260)-Net over F₃₂ — Constructive and digital

Digital (57, 105, 260)-net over F₃₂, using

1 times m-reduction [i] based on digital (57, 106, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 19, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (7, 31, 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]
  3. digital (7, 56, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(57, 57+48, 513)-Net in Base 32 — Constructive

(57, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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+48, 1389)-Net over F₃₂ — Digital

Digital (57, 105, 1389)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+48, 1216263)-Net in Base 32 — Upper bound on s

There is no (57, 105, 1216264)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 836990 145872 655404 883019 696082 519207 854059 557851 946997 278202 699956 486503 802664 004055 553297 671988 171038 347490 561230 953015 280979 139479 579094 144396 384790 247968 > 32¹⁰⁵ [i]