MinT - Best Known (52, 52+57, s)-Nets in Base 27

Best Known (52, 52+57, s)-Nets in Base 27

(52, 52+57, 192)-Net over F₂₇ — Constructive and digital

Digital (52, 109, 192)-net over F₂₇, using

t-expansion [i] based on digital (51, 109, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]
  2. digital (11, 69, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(52, 52+57, 370)-Net in Base 27 — Constructive

(52, 109, 370)-net in base 27, using

27¹ times duplication [i] based on (51, 108, 370)-net in base 27, using
- t-expansion [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(52, 52+57, 507)-Net over F₂₇ — Digital

Digital (52, 109, 507)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+57, 144199)-Net in Base 27 — Upper bound on s

There is no (52, 109, 144200)-net in base 27, because

1 times m-reduction [i] would yield (52, 108, 144200)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38669 573249 299109 143150 467447 785557 002762 782318 458585 039305 170476 548586 187601 459874 587520 399150 448460 991075 970134 090809 091832 982289 569143 774503 735795 611969 > 27¹⁰⁸ [i]