MinT - Best Known (77−32, 77, s)-Nets in Base 32

Best Known (77−32, 77, s)-Nets in Base 32

(77−32, 77, 272)-Net over F₃₂ — Constructive and digital

Digital (45, 77, 272)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 15, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 23, 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]
3. digital (7, 39, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(77−32, 77, 514)-Net in Base 32 — Constructive

(45, 77, 514)-net in base 32, using

base change [i] based on (23, 55, 514)-net in base 128, using
- 1 times m-reduction [i] based on (23, 56, 514)-net in base 128, using
  - base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 16, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
      2. digital (0, 33, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(77−32, 77, 2210)-Net over F₃₂ — Digital

Digital (45, 77, 2210)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−32, 77, 3843354)-Net in Base 32 — Upper bound on s

There is no (45, 77, 3843355)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 804317 986885 644839 644206 883178 600907 579367 824729 930148 388745 393402 208759 671929 179942 475165 861671 045603 023253 691463 > 32⁷⁷ [i]