MinT - Best Known (45, 78, s)-Nets in Base 32

Best Known (45, 78, s)-Nets in Base 32

(45, 78, 260)-Net over F₃₂ — Constructive and digital

Digital (45, 78, 260)-net over F₃₂, using

2 times m-reduction [i] based on digital (45, 80, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 14, 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, 24, 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, 42, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(45, 78, 513)-Net in Base 32 — Constructive

(45, 78, 513)-net in base 32, using

24 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(45, 78, 1941)-Net over F₃₂ — Digital

Digital (45, 78, 1941)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 78, 3843354)-Net in Base 32 — Upper bound on s

There is no (45, 78, 3843355)-net in base 32, because

1 times m-reduction [i] would yield (45, 77, 3843355)-net in base 32, but
- 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]