MinT - Best Known (176−32, 176, s)-Nets in Base 4

Best Known (176−32, 176, s)-Nets in Base 4

(176−32, 176, 1223)-Net over F₄ — Constructive and digital

Digital (144, 176, 1223)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (32, 48, 195)-net over F₄, using
  - trace code for nets [i] based on digital (0, 16, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (96, 128, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 32, 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]

(176−32, 176, 13028)-Net over F₄ — Digital

Digital (144, 176, 13028)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁶, 13028, F₄, 32) (dual of [13028, 12852, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷⁶, 16420, F₄, 32) (dual of [16420, 16244, 33]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(4¹⁶⁹, 16385, F₄, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹⁴¹, 16385, F₄, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [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]

(176−32, 176, large)-Net in Base 4 — Upper bound on s

There is no (144, 176, large)-net in base 4, because

30 times m-reduction [i] would yield (144, 146, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]