MinT - Best Known (58, 105, s)-Nets in Base 32

Best Known (58, 105, s)-Nets in Base 32

(58, 105, 272)-Net over F₃₂ — Constructive and digital

Digital (58, 105, 272)-net over F₃₂, using

32¹ times duplication [i] based on digital (57, 104, 272)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (5, 20, 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, 30, 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, 54, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(58, 105, 513)-Net in Base 32 — Constructive

(58, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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]

(58, 105, 1606)-Net over F₃₂ — Digital

Digital (58, 105, 1606)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 105, 1945304)-Net in Base 32 — Upper bound on s

There is no (58, 105, 1945305)-net in base 32, because

1 times m-reduction [i] would yield (58, 104, 1945305)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 432432 396841 986695 305571 207784 015885 531076 349887 309175 061994 951370 052506 375073 487160 043225 610581 605872 084377 552822 298986 876621 050480 565014 724246 162856 595656 > 32¹⁰⁴ [i]