MinT - Best Known (54, 54+14, s)-Nets in Base 4

Best Known (54, 54+14, s)-Nets in Base 4

(54, 54+14, 1048)-Net over F₄ — Constructive and digital

Digital (54, 68, 1048)-net over F₄, using

4¹ times duplication [i] based on digital (53, 67, 1048)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 11, 20)-net over F₄, using
    - linear code embedding found in a computer search [i]
  2. digital (42, 56, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 14, 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]

(54, 54+14, 4044)-Net over F₄ — Digital

Digital (54, 68, 4044)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶⁸, 4044, F₄, 14) (dual of [4044, 3976, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶⁸, 4104, F₄, 14) (dual of [4104, 4036, 15]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁷, 8, F₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
      - dual of repetition code with length 8 [i]
    2. linear OA(4⁶¹, 4096, F₄, 14) (dual of [4096, 4035, 15]-code), using
      - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(54, 54+14, 795018)-Net in Base 4 — Upper bound on s

There is no (54, 68, 795019)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 87112 644075 047532 272906 605529 743758 476320 > 4⁶⁸ [i]