MinT - Best Known (196, 211, s)-Nets in Base 4

Best Known (196, 211, s)-Nets in Base 4

(196, 211, 4798952)-Net over F₄ — Constructive and digital

Digital (196, 211, 4798952)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (32, 39, 5468)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 5)-net over F₄, using
      - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
        the rational function field F₄(x) [i]
        Niederreiter sequence [i]
    2. digital (29, 36, 5463)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4³⁶, 5463, F₄, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(4³⁶, 16390, F₄, 7) (dual of [16390, 16354, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(4³⁶, 16391, F₄, 7) (dual of [16391, 16355, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(4³⁶, 16384, F₄, 7) (dual of [16384, 16348, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(4²⁹, 16384, F₄, 6) (dual of [16384, 16355, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (157, 172, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(196, 211, large)-Net over F₄ — Digital

Digital (196, 211, large)-net over F₄, using

t-expansion [i] based on digital (195, 211, large)-net over F₄, using
- 6 times m-reduction [i] based on digital (195, 217, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁷, large, F₄, 22) (dual of [large, large−217, 23]-code), using
    - strength reduction [i] based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(196, 211, large)-Net in Base 4 — Upper bound on s

There is no (196, 211, large)-net in base 4, because

13 times m-reduction [i] would yield (196, 198, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (196, 211, s)-Nets in Base 4

(196, 211, 4798952)-Net over F4 — Constructive and digital

(196, 211, large)-Net over F4 — Digital

(196, 211, large)-Net in Base 4 — Upper bound on s

(196, 211, 4798952)-Net over F₄ — Constructive and digital

(196, 211, large)-Net over F₄ — Digital