MinT - Best Known (56, 56+43, s)-Nets in Base 32

Best Known (56, 56+43, s)-Nets in Base 32

(56, 56+43, 294)-Net over F₃₂ — Constructive and digital

Digital (56, 99, 294)-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 (7, 28, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 50, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(56, 56+43, 513)-Net in Base 32 — Constructive

(56, 99, 513)-net in base 32, using

t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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]

(56, 56+43, 1902)-Net over F₃₂ — Digital

Digital (56, 99, 1902)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+43, 2959119)-Net in Base 32 — Upper bound on s

There is no (56, 99, 2959120)-net in base 32, because

1 times m-reduction [i] would yield (56, 98, 2959120)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3196 676513 946901 276247 131808 279205 504357 557234 136635 841708 280035 291039 726901 757878 612427 497801 241375 912409 197213 574681 771820 634249 633877 510761 633072 > 32⁹⁸ [i]