MinT - Best Known (106−44, 106, s)-Nets in Base 32

Best Known (106−44, 106, s)-Nets in Base 32

(106−44, 106, 316)-Net over F₃₂ — Constructive and digital

Digital (62, 106, 316)-net over F₃₂, using

1 times m-reduction [i] based on digital (62, 107, 316)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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, 29, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (11, 56, 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]

(106−44, 106, 513)-Net in Base 32 — Constructive

(62, 106, 513)-net in base 32, using

t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 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]

(106−44, 106, 2817)-Net over F₃₂ — Digital

Digital (62, 106, 2817)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−44, 106, 5218872)-Net in Base 32 — Upper bound on s

There is no (62, 106, 5218873)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3514 788702 655672 224645 282131 700875 092265 375180 343646 844721 384078 099001 987624 508903 968328 254061 036454 300340 354762 930204 216727 580965 863521 488088 397462 665848 222428 > 32¹⁰⁶ [i]