MinT - Best Known (89−53, 89, s)-Nets in Base 32

Best Known (89−53, 89, s)-Nets in Base 32

(89−53, 89, 162)-Net over F₃₂ — Constructive and digital

Digital (36, 89, 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, 60, 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]

(89−53, 89, 273)-Net over F₃₂ — Digital

Digital (36, 89, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 89, 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]

(89−53, 89, 288)-Net in Base 32 — Constructive

(36, 89, 288)-net in base 32, using

t-expansion [i] based on (35, 89, 288)-net in base 32, using
- 2 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]

(89−53, 89, 342)-Net in Base 32

(36, 89, 342)-net in base 32, using

7 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]

(89−53, 89, 42281)-Net in Base 32 — Upper bound on s

There is no (36, 89, 42282)-net in base 32, because

1 times m-reduction [i] would yield (36, 88, 42282)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 840468 353222 968372 935909 744137 054158 654239 699756 454783 125068 196351 020463 850522 240771 728944 510652 892112 666215 336546 123890 871618 049744 > 32⁸⁸ [i]