MinT - Best Known (185−34, 185, s)-Nets in Base 4

Best Known (185−34, 185, s)-Nets in Base 4

(185−34, 185, 1158)-Net over F₄ — Constructive and digital

Digital (151, 185, 1158)-net over F₄, using

1 times m-reduction [i] based on digital (151, 186, 1158)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (29, 46, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 23, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (105, 140, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 35, 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]

(185−34, 185, 12323)-Net over F₄ — Digital

Digital (151, 185, 12323)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁵, 12323, F₄, 34) (dual of [12323, 12138, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸⁵, 16421, F₄, 34) (dual of [16421, 16236, 35]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(4¹⁸³, 16419, F₄, 34) (dual of [16419, 16236, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(28) [i] based on
      1. linear OA(4¹⁷⁶, 16384, F₄, 34) (dual of [16384, 16208, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(4¹⁴⁸, 16384, F₄, 29) (dual of [16384, 16236, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(185−34, 185, large)-Net in Base 4 — Upper bound on s

There is no (151, 185, large)-net in base 4, because

32 times m-reduction [i] would yield (151, 153, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]