MinT - Best Known (110−49, 110, s)-Nets in Base 32

Best Known (110−49, 110, s)-Nets in Base 32

(110−49, 110, 294)-Net over F₃₂ — Constructive and digital

Digital (61, 110, 294)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 23, 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]
2. digital (7, 31, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 56, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(110−49, 110, 513)-Net in Base 32 — Constructive

(61, 110, 513)-net in base 32, using

32² 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]

(110−49, 110, 1725)-Net over F₃₂ — Digital

Digital (61, 110, 1725)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−49, 110, 2167145)-Net in Base 32 — Upper bound on s

There is no (61, 110, 2167146)-net in base 32, because

1 times m-reduction [i] would yield (61, 109, 2167146)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 172848 933807 554327 009078 257991 341177 314835 850024 042948 647666 775647 096832 869544 266022 014727 184433 274749 240637 826280 895114 447982 048937 680378 109400 348754 625592 689846 > 32¹⁰⁹ [i]