MinT - Best Known (200−14, 200, s)-Nets in Base 4

Best Known (200−14, 200, s)-Nets in Base 4

(200−14, 200, 4798964)-Net over F₄ — Constructive and digital

Digital (186, 200, 4798964)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 40, 5480)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [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 (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]

(200−14, 200, large)-Net over F₄ — Digital

Digital (186, 200, large)-net over F₄, using

7 times m-reduction [i] based on digital (186, 207, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁷, large, F₄, 21) (dual of [large, large−207, 22]-code), using
  - 26 times code embedding in larger space [i] based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(200−14, 200, large)-Net in Base 4 — Upper bound on s

There is no (186, 200, large)-net in base 4, because

12 times m-reduction [i] would yield (186, 188, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (200−14, 200, s)-Nets in Base 4

(200−14, 200, 4798964)-Net over F4 — Constructive and digital

(200−14, 200, large)-Net over F4 — Digital

(200−14, 200, large)-Net in Base 4 — Upper bound on s

(200−14, 200, 4798964)-Net over F₄ — Constructive and digital

(200−14, 200, large)-Net over F₄ — Digital