MinT - Best Known (109−66, 109, s)-Nets in Base 32

Best Known (109−66, 109, s)-Nets in Base 32

(109−66, 109, 162)-Net over F₃₂ — Constructive and digital

Digital (43, 109, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 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, 73, 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]

(109−66, 109, 288)-Net in Base 32 — Constructive

(43, 109, 288)-net in base 32, using

32¹ times duplication [i] based on (42, 108, 288)-net in base 32, using
- t-expansion [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]

(109−66, 109, 308)-Net over F₃₂ — Digital

Digital (43, 109, 308)-net over F₃₂, using

t-expansion [i] based on digital (41, 109, 308)-net over F₃₂, using
- net from sequence [i] based on digital (41, 307)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 41 and N(F) ≥ 308, using
    - a curve from the manYPoints database [i]

(109−66, 109, 342)-Net in Base 32

(43, 109, 342)-net in base 32, using

32¹ times duplication [i] based on (42, 108, 342)-net in base 32, using
- t-expansion [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]

(109−66, 109, 39752)-Net in Base 32 — Upper bound on s

There is no (43, 109, 39753)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 241099 031399 525985 367917 449345 918152 352860 909160 155644 874193 988485 562004 763074 179119 901003 240024 729483 713982 226109 128808 982306 474985 119597 081913 797366 200942 866512 > 32¹⁰⁹ [i]