MinT - Best Known (49, 104, s)-Nets in Base 32

Best Known (49, 104, s)-Nets in Base 32

(49, 104, 240)-Net over F₃₂ — Constructive and digital

Digital (49, 104, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 38, 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, 66, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(49, 104, 513)-Net in Base 32 — Constructive

(49, 104, 513)-net in base 32, using

t-expansion [i] based on (46, 104, 513)-net in base 32, using
- 4 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]

(49, 104, 537)-Net over F₃₂ — Digital

Digital (49, 104, 537)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 104, 194483)-Net in Base 32 — Upper bound on s

There is no (49, 104, 194484)-net in base 32, because

1 times m-reduction [i] would yield (49, 103, 194484)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107271 394732 333401 679093 314726 239180 152629 542889 567676 417194 996684 907659 230501 920035 567067 174400 148043 737947 930490 522012 123571 866221 557973 780120 439789 343844 > 32¹⁰³ [i]