MinT - Best Known (87−51, 87, s)-Nets in Base 32

Best Known (87−51, 87, s)-Nets in Base 32

(87−51, 87, 162)-Net over F₃₂ — Constructive and digital

Digital (36, 87, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (36, 88, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 29, 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, 59, 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]

(87−51, 87, 273)-Net over F₃₂ — Digital

Digital (36, 87, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 87, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(87−51, 87, 288)-Net in Base 32 — Constructive

(36, 87, 288)-net in base 32, using

t-expansion [i] based on (35, 87, 288)-net in base 32, using
- 4 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
  - base change [i] based on digital (9, 65, 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]

(87−51, 87, 342)-Net in Base 32

(36, 87, 342)-net in base 32, using

9 times m-reduction [i] based on (36, 96, 342)-net in base 32, using
- base change [i] based on digital (20, 80, 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]

(87−51, 87, 49415)-Net in Base 32 — Upper bound on s

There is no (36, 87, 49416)-net in base 32, because

1 times m-reduction [i] would yield (36, 86, 49416)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2772 679050 491664 411301 691888 732030 067384 165311 391508 891302 429997 754216 212025 962804 082385 354822 637917 960547 488714 840909 705221 443808 > 32⁸⁶ [i]