MinT - Best Known (196−12, 196, s)-Nets in Base 4

Best Known (196−12, 196, s)-Nets in Base 4

(196−12, 196, large)-Net over F₄ — Constructive and digital

Digital (184, 196, large)-net over F₄, using

4⁸ times duplication [i] based on digital (176, 188, large)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (46, 52, 2796206)-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 (43, 49, 2796201)-net over F₄, using
        net defined by OOA [i] based on linear OOA(4⁴⁹, 2796201, F₄, 6, 6) (dual of [(2796201, 6), 16777157, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(4⁴⁹, large, F₄, 6) (dual of [large, large−49, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. digital (124, 136, 5592400)-net over F₄, using
    - trace code for nets [i] based on digital (56, 68, 2796200)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(196−12, 196, large)-Net in Base 4 — Upper bound on s

There is no (184, 196, large)-net in base 4, because

10 times m-reduction [i] would yield (184, 186, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (196−12, 196, s)-Nets in Base 4

(196−12, 196, large)-Net over F4 — Constructive and digital

(196−12, 196, large)-Net in Base 4 — Upper bound on s

(196−12, 196, large)-Net over F₄ — Constructive and digital