MinT - Best Known (45, 56, s)-Nets in Base 4

Best Known (45, 56, s)-Nets in Base 4

(45, 56, 1109)-Net over F₄ — Constructive and digital

Digital (45, 56, 1109)-net over F₄, using

4¹ times duplication [i] based on digital (44, 55, 1109)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 11, 81)-net over F₄, using
    - linear code embedding found in a computer search [i]
  2. digital (33, 44, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 11, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(45, 56, 5461)-Net over F₄ — Digital

Digital (45, 56, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁶, 5461, F₄, 11) (dual of [5461, 5405, 12]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,31}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(45, 56, 3642289)-Net in Base 4 — Upper bound on s

There is no (45, 56, 3642290)-net in base 4, because

1 times m-reduction [i] would yield (45, 55, 3642290)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1298 075104 873610 074291 539410 460220 > 4⁵⁵ [i]