MinT - Best Known (214, 214+14, s)-Nets in Base 3

Best Known (214, 214+14, s)-Nets in Base 3

(214, 214+14, 4793518)-Net over F₃ — Constructive and digital

Digital (214, 228, 4793518)-net over F₃, using

3² times duplication [i] based on digital (212, 226, 4793518)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 14, 34)-net over F₃, using
    - linear code embedding found in a computer search [i]
  2. digital (198, 212, 4793484)-net over F₃, using
    - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(214, 214+14, large)-Net over F₃ — Digital

Digital (214, 228, large)-net over F₃, using

t-expansion [i] based on digital (209, 228, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, large, F₃, 19) (dual of [large, large−228, 20]-code), using
  - 47 times code embedding in larger space [i] based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(214, 214+14, large)-Net in Base 3 — Upper bound on s

There is no (214, 228, large)-net in base 3, because

12 times m-reduction [i] would yield (214, 216, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]