MinT - Best Known (208−12, 208, s)-Nets in Base 3

Best Known (208−12, 208, s)-Nets in Base 3

(208−12, 208, 5593129)-Net over F₃ — Constructive and digital

Digital (196, 208, 5593129)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (22, 28, 729)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁸, 729, F₃, 6, 6) (dual of [(729, 6), 4346, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3²⁸, 729, F₃, 5, 6) (dual of [(729, 5), 3617, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3²⁸, 2187, F₃, 6) (dual of [2187, 2159, 7]-code), using
        1 times truncation [i] based on linear OA(3²⁹, 2188, F₃, 7) (dual of [2188, 2159, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
2. digital (168, 180, 5592400)-net over F₃, using
  - trace code for nets [i] based on digital (78, 90, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁰, 2796200, F₉, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁰, 8388601, F₉, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁰, 8388602, F₉, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁵, 4194301, F₈₁, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁵, 8388602, F₈₁, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(208−12, 208, large)-Net over F₃ — Digital

Digital (196, 208, large)-net over F₃, using

3⁵ times duplication [i] based on digital (191, 203, large)-net over F₃, using
- t-expansion [i] based on digital (186, 203, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰³, large, F₃, 17) (dual of [large, large−203, 18]-code), using
    - 37 times code embedding in larger space [i] based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(208−12, 208, large)-Net in Base 3 — Upper bound on s

There is no (196, 208, large)-net in base 3, because

10 times m-reduction [i] would yield (196, 198, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (208−12, 208, s)-Nets in Base 3

(208−12, 208, 5593129)-Net over F3 — Constructive and digital

(208−12, 208, large)-Net over F3 — Digital

(208−12, 208, large)-Net in Base 3 — Upper bound on s

(208−12, 208, 5593129)-Net over F₃ — Constructive and digital

(208−12, 208, large)-Net over F₃ — Digital