MinT - Best Known (172−33, 172, s)-Nets in Base 4

Best Known (172−33, 172, s)-Nets in Base 4

(172−33, 172, 1118)-Net over F₄ — Constructive and digital

Digital (139, 172, 1118)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (24, 40, 90)-net over F₄, using
  - trace code for nets [i] based on digital (4, 20, 45)-net over F₁₆, using
    - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
        a function field by GarcÃa and Quoos [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]

(172−33, 172, 8644)-Net over F₄ — Digital

Digital (139, 172, 8644)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷², 8644, F₄, 33) (dual of [8644, 8472, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷², 16391, F₄, 33) (dual of [16391, 16219, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [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(4¹⁵⁵, 16385, F₄, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(4³, 6, F₄, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
      - hyperoval in PG(2,Â 4) [i]
      - the Hexacode [i]

(172−33, 172, 6164994)-Net in Base 4 — Upper bound on s

There is no (139, 172, 6164995)-net in base 4, because

1 times m-reduction [i] would yield (139, 171, 6164995)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 958980 797921 001977 337161 285390 270990 638154 504440 492841 297721 393855 457429 646736 112196 980073 586057 554912 > 4¹⁷¹ [i]