MinT - Best Known (194−14, 194, s)-Nets in Base 4

Best Known (194−14, 194, s)-Nets in Base 4

(194−14, 194, 4794856)-Net over F₄ — Constructive and digital

Digital (180, 194, 4794856)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (27, 34, 1372)-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 (24, 31, 1367)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4³¹, 1367, F₄, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(4³¹, 4102, F₄, 7) (dual of [4102, 4071, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(4³¹, 4096, F₄, 7) (dual of [4096, 4065, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(4²⁵, 4096, F₄, 6) (dual of [4096, 4071, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 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 (146, 160, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(194−14, 194, large)-Net over F₄ — Digital

Digital (180, 194, large)-net over F₄, using

t-expansion [i] based on digital (177, 194, large)-net over F₄, using
- 3 times m-reduction [i] based on digital (177, 197, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁷, large, F₄, 20) (dual of [large, large−197, 21]-code), using
    - 17 times code embedding in larger space [i] based on linear OA(4¹⁸⁰, large, F₄, 20) (dual of [large, large−180, 21]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(194−14, 194, large)-Net in Base 4 — Upper bound on s

There is no (180, 194, large)-net in base 4, because

12 times m-reduction [i] would yield (180, 182, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (194−14, 194, s)-Nets in Base 4

(194−14, 194, 4794856)-Net over F4 — Constructive and digital

(194−14, 194, large)-Net over F4 — Digital

(194−14, 194, large)-Net in Base 4 — Upper bound on s

(194−14, 194, 4794856)-Net over F₄ — Constructive and digital

(194−14, 194, large)-Net over F₄ — Digital