MinT - Best Known (108−55, 108, s)-Nets in Base 32

Best Known (108−55, 108, s)-Nets in Base 32

(108−55, 108, 240)-Net over F₃₂ — Constructive and digital

Digital (53, 108, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 108, 240)-net over F₃₂, using
- 1 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)

(108−55, 108, 513)-Net in Base 32 — Constructive

(53, 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]

(108−55, 108, 703)-Net over F₃₂ — Digital

Digital (53, 108, 703)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−55, 108, 324997)-Net in Base 32 — Upper bound on s

There is no (53, 108, 324998)-net in base 32, because

1 times m-reduction [i] would yield (53, 107, 324998)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 112476 548055 301218 256996 609200 322776 857039 658477 899629 506000 924115 496817 399719 108446 760084 295561 610840 864863 726944 185806 097537 425282 428342 440005 703964 900761 385888 > 32¹⁰⁷ [i]