MinT - Best Known (53, 94, s)-Nets in Base 32

Best Known (53, 94, s)-Nets in Base 32

(53, 94, 272)-Net over F₃₂ — Constructive and digital

Digital (53, 94, 272)-net over F₃₂, using

32¹ times duplication [i] based on digital (52, 93, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 18, 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, 27, 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, 48, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(53, 94, 513)-Net in Base 32 — Constructive

(53, 94, 513)-net in base 32, using

t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 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, 94, 1772)-Net over F₃₂ — Digital

Digital (53, 94, 1772)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 94, 2672331)-Net in Base 32 — Upper bound on s

There is no (53, 94, 2672332)-net in base 32, because

1 times m-reduction [i] would yield (53, 93, 2672332)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 95 268752 503535 209099 682540 812168 720863 445318 118981 247316 358989 838851 118464 850714 818578 232661 178154 876353 398328 045672 043720 923428 160940 348168 > 32⁹³ [i]