MinT - Best Known (90−48, 90, s)-Nets in Base 32

Best Known (90−48, 90, s)-Nets in Base 32

(90−48, 90, 218)-Net over F₃₂ — Constructive and digital

Digital (42, 90, 218)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (11, 59, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(90−48, 90, 288)-Net in Base 32 — Constructive

(42, 90, 288)-net in base 32, using

t-expansion [i] based on (40, 90, 288)-net in base 32, using
- 18 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]

(90−48, 90, 455)-Net over F₃₂ — Digital

Digital (42, 90, 455)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−48, 90, 513)-Net in Base 32

(42, 90, 513)-net in base 32, using

base change [i] based on (27, 75, 513)-net in base 64, using
- 1 times m-reduction [i] based on (27, 76, 513)-net in base 64, using
  - base change [i] based on digital (8, 57, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(90−48, 90, 139403)-Net in Base 32 — Upper bound on s

There is no (42, 90, 139404)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 356578 379995 286422 678625 960568 552048 649703 957894 081930 382438 368573 846901 194143 169285 475582 453612 494776 300216 529382 684677 470144 504552 > 32⁹⁰ [i]