MinT - Best Known (110−51, 110, s)-Nets in Base 32

Best Known (110−51, 110, s)-Nets in Base 32

(110−51, 110, 260)-Net over F₃₂ — Constructive and digital

Digital (59, 110, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 20, 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, 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, 58, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(110−51, 110, 513)-Net in Base 32 — Constructive

(59, 110, 513)-net in base 32, using

32² times duplication [i] based on (57, 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]

(110−51, 110, 1312)-Net over F₃₂ — Digital

Digital (59, 110, 1312)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−51, 110, 1198707)-Net in Base 32 — Upper bound on s

There is no (59, 110, 1198708)-net in base 32, because

1 times m-reduction [i] would yield (59, 109, 1198708)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 172224 502485 608135 251133 755546 188277 123895 673155 239805 129571 284735 605836 475973 164354 166353 507264 416071 700828 264832 528402 950424 939570 261857 581288 810697 069321 826500 > 32¹⁰⁹ [i]