MinT - Best Known (106−53, 106, s)-Nets in Base 32

Best Known (106−53, 106, s)-Nets in Base 32

(106−53, 106, 240)-Net over F₃₂ — Constructive and digital

Digital (53, 106, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 106, 240)-net over F₃₂, using
- 3 times m-reduction [i] based on digital (51, 109, 240)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 40, 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]
    2. digital (11, 69, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(106−53, 106, 513)-Net in Base 32 — Constructive

(53, 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]

(106−53, 106, 781)-Net over F₃₂ — Digital

Digital (53, 106, 781)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−53, 106, 407762)-Net in Base 32 — Upper bound on s

There is no (53, 106, 407763)-net in base 32, because

1 times m-reduction [i] would yield (53, 105, 407763)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 839256 999869 852018 886376 140558 428600 413050 963380 782557 011777 716952 027398 814730 165186 585309 640058 007439 975849 451925 736175 058832 064034 632547 365600 032104 189528 > 32¹⁰⁵ [i]