MinT - Best Known (53, 53+42, s)-Nets in Base 32

Best Known (53, 53+42, s)-Nets in Base 32

(53, 53+42, 260)-Net over F₃₂ — Constructive and digital

Digital (53, 95, 260)-net over F₃₂, using

2 times m-reduction [i] based on digital (53, 97, 260)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (3, 17, 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, 29, 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, 51, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(53, 53+42, 513)-Net in Base 32 — Constructive

(53, 95, 513)-net in base 32, using

t-expansion [i] based on (46, 95, 513)-net in base 32, using
- 13 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 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]

(53, 53+42, 1620)-Net over F₃₂ — Digital

Digital (53, 95, 1620)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+42, 1803599)-Net in Base 32 — Upper bound on s

There is no (53, 95, 1803600)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 97555 196949 865208 515832 412418 030725 531630 728718 837604 770286 913798 127736 595654 173826 667219 728282 517061 511039 069020 607949 070985 947542 999117 538756 > 32⁹⁵ [i]