MinT - Best Known (105−65, 105, s)-Nets in Base 32

Best Known (105−65, 105, s)-Nets in Base 32

(105−65, 105, 142)-Net over F₃₂ — Constructive and digital

Digital (40, 105, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 33, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 72, 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]

(105−65, 105, 288)-Net in Base 32 — Constructive

(40, 105, 288)-net in base 32, using

3 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
- base change [i] based on (22, 90, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(105−65, 105, 293)-Net over F₃₂ — Digital

Digital (40, 105, 293)-net over F₃₂, using

net from sequence [i] based on digital (40, 292)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 40 and N(F) ≥ 293, using
  - a curve from the manYPoints database [i]

(105−65, 105, 342)-Net in Base 32

(40, 105, 342)-net in base 32, using

t-expansion [i] based on (38, 105, 342)-net in base 32, using
- 3 times m-reduction [i] based on (38, 108, 342)-net in base 32, using
  - base change [i] based on digital (20, 90, 342)-net over F₆₄, using
    - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
        a curve from the manYPoints database [i]

(105−65, 105, 32139)-Net in Base 32 — Upper bound on s

There is no (40, 105, 32140)-net in base 32, because

1 times m-reduction [i] would yield (40, 104, 32140)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 435708 694582 057652 659721 271597 778704 205284 926085 137454 341846 038126 857124 490860 207620 219006 276092 669032 332047 247067 892996 712370 139076 465651 031446 156262 397348 > 32¹⁰⁴ [i]