MinT - Best Known (109−49, 109, s)-Nets in Base 32

Best Known (109−49, 109, s)-Nets in Base 32

(109−49, 109, 272)-Net over F₃₂ — Constructive and digital

Digital (60, 109, 272)-net over F₃₂, using

1 times m-reduction [i] based on digital (60, 110, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 21, 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, 32, 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, 57, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(109−49, 109, 513)-Net in Base 32 — Constructive

(60, 109, 513)-net in base 32, using

32¹ times duplication [i] based on (59, 108, 513)-net in base 32, using
- t-expansion [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]

(109−49, 109, 1607)-Net over F₃₂ — Digital

Digital (60, 109, 1607)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−49, 109, 1875742)-Net in Base 32 — Upper bound on s

There is no (60, 109, 1875743)-net in base 32, because

1 times m-reduction [i] would yield (60, 108, 1875743)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 599177 059152 848325 308377 864581 078836 954309 330091 788372 047474 049360 232741 782785 719973 824007 810760 214530 669489 602981 639776 420890 216102 289936 961536 685768 918705 068223 > 32¹⁰⁸ [i]