MinT - Best Known (52, 52+53, s)-Nets in Base 32

Best Known (52, 52+53, s)-Nets in Base 32

(52, 52+53, 240)-Net over F₃₂ — Constructive and digital

Digital (52, 105, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 105, 240)-net over F₃₂, using
- 4 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)

(52, 52+53, 513)-Net in Base 32 — Constructive

(52, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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]

(52, 52+53, 728)-Net over F₃₂ — Digital

Digital (52, 105, 728)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+53, 356874)-Net in Base 32 — Upper bound on s

There is no (52, 105, 356875)-net in base 32, because

1 times m-reduction [i] would yield (52, 104, 356875)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 432624 584427 328047 543770 738109 586603 576960 531457 362735 350544 387416 621510 432996 532792 309450 537396 771896 147496 741027 090018 098590 114542 299023 632691 521582 431776 > 32¹⁰⁴ [i]