MinT - Best Known (54, 54+56, s)-Nets in Base 27

Best Known (54, 54+56, s)-Nets in Base 27

(54, 54+56, 192)-Net over F₂₇ — Constructive and digital

Digital (54, 110, 192)-net over F₂₇, using

t-expansion [i] based on digital (51, 110, 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, 70, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(54, 54+56, 370)-Net in Base 27 — Constructive

(54, 110, 370)-net in base 27, using

27² times duplication [i] based on (52, 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]

(54, 54+56, 601)-Net over F₂₇ — Digital

Digital (54, 110, 601)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+56, 182478)-Net in Base 27 — Upper bound on s

There is no (54, 110, 182479)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 28 186765 538481 636313 185626 424942 445239 030163 634958 152562 611109 085899 995329 933813 113931 137628 450253 273736 581220 586291 044264 004584 082349 093835 189543 844157 389353 > 27¹¹⁰ [i]