MinT - Best Known (179−33, 179, s)-Nets in Base 4

Best Known (179−33, 179, s)-Nets in Base 4

(179−33, 179, 1158)-Net over F₄ — Constructive and digital

Digital (146, 179, 1158)-net over F₄, using

4³ times duplication [i] based on digital (143, 176, 1158)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 44, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 22, 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 (99, 132, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 33, 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]

(179−33, 179, 11830)-Net over F₄ — Digital

Digital (146, 179, 11830)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁹, 11830, F₄, 33) (dual of [11830, 11651, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷⁹, 16423, F₄, 33) (dual of [16423, 16244, 34]-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(4¹⁰, 38, F₄, 5) (dual of [38, 28, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(179−33, 179, large)-Net in Base 4 — Upper bound on s

There is no (146, 179, large)-net in base 4, because

31 times m-reduction [i] would yield (146, 148, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]