MinT - Best Known (109−44, 109, s)-Nets in Base 32

Best Known (109−44, 109, s)-Nets in Base 32

(109−44, 109, 338)-Net over F₃₂ — Constructive and digital

Digital (65, 109, 338)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 21, 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 (11, 33, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]
3. digital (11, 55, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(109−44, 109, 546)-Net in Base 32 — Constructive

(65, 109, 546)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. (43, 87, 513)-net in base 32, using
  - 3 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
    - base change [i] based on digital (28, 75, 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]

(109−44, 109, 3582)-Net over F₃₂ — Digital

Digital (65, 109, 3582)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−44, 109, 8371913)-Net in Base 32 — Upper bound on s

There is no (65, 109, 8371914)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 172329 309209 600224 513733 006052 477797 261868 099045 296948 116296 730954 226411 037512 877969 666316 682868 992634 295421 360298 911450 340401 924440 659442 388666 801933 331683 339216 > 32¹⁰⁹ [i]