MinT - Best Known (90−55, 90, s)-Nets in Base 32

Best Known (90−55, 90, s)-Nets in Base 32

(90−55, 90, 142)-Net over F₃₂ — Constructive and digital

Digital (35, 90, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 28, 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, 62, 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]

(90−55, 90, 273)-Net over F₃₂ — Digital

Digital (35, 90, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 90, 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]

(90−55, 90, 288)-Net in Base 32 — Constructive

(35, 90, 288)-net in base 32, using

1 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]

(90−55, 90, 342)-Net in Base 32

(35, 90, 342)-net in base 32, using

base change [i] based on digital (20, 75, 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]

(90−55, 90, 32231)-Net in Base 32 — Upper bound on s

There is no (35, 90, 32232)-net in base 32, because

1 times m-reduction [i] would yield (35, 89, 32232)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 904103 422879 788111 481936 261749 016697 721935 776348 517410 334399 822027 065319 997413 719516 492998 373009 694476 951986 118258 692375 581111 365228 > 32⁸⁹ [i]