MinT - Best Known (45, 45+47, s)-Nets in Base 32

Best Known (45, 45+47, s)-Nets in Base 32

(45, 45+47, 240)-Net over F₃₂ — Constructive and digital

Digital (45, 92, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 34, 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]
2. digital (11, 58, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(45, 45+47, 513)-Net in Base 32 — Constructive

(45, 92, 513)-net in base 32, using

10 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, 45+47, 609)-Net over F₃₂ — Digital

Digital (45, 92, 609)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+47, 274305)-Net in Base 32 — Upper bound on s

There is no (45, 92, 274306)-net in base 32, because

1 times m-reduction [i] would yield (45, 91, 274306)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 93038 813059 337231 227348 205026 766965 730578 374061 304600 556032 240852 632886 747133 197840 273157 870097 929963 601642 891492 077059 686329 967336 676544 > 32⁹¹ [i]