MinT - Best Known (223, 223+12, s)-Nets in Base 3

Best Known (223, 223+12, s)-Nets in Base 3

(223, 223+12, 6990613)-Net over F₃ — Constructive and digital

Digital (223, 235, 6990613)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (61, 67, 2796313)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁶⁷, 2796313, F₃, 6, 6) (dual of [(2796313, 6), 16777811, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁶⁷, 2796313, F₃, 5, 6) (dual of [(2796313, 5), 13981498, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3⁷, 112, F₃, 5, 3) (dual of [(112, 5), 553, 4]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁷, 112, F₃, 3, 3) (dual of [(112, 3), 329, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(3⁷, 112, F₃, 2, 3) (dual of [(112, 2), 217, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁷, 112, F₃, 3) (dual of [112, 105, 4]-code or 112-cap in PG(6,3)), using
        doubling the cap [i] based on linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
        the Hill cap [i]
        linear OOA(3⁶⁰, 2796201, F₃, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
2. digital (156, 168, 4194300)-net over F₃, using
  - trace code for nets [i] based on digital (44, 56, 1398100)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁵⁶, 1398100, F₂₇, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(27⁵⁶, 8388600, F₂₇, 12) (dual of [8388600, 8388544, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵⁶, large, F₂₇, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(223, 223+12, large)-Net over F₃ — Digital

Digital (223, 235, large)-net over F₃, using

t-expansion [i] based on digital (221, 235, large)-net over F₃, using
- 6 times m-reduction [i] based on digital (221, 241, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, large, F₃, 20) (dual of [large, large−241, 21]-code), using
    - strength reduction [i] based on linear OA(3²⁴¹, large, F₃, 25) (dual of [large, large−241, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(223, 223+12, large)-Net in Base 3 — Upper bound on s

There is no (223, 235, large)-net in base 3, because

10 times m-reduction [i] would yield (223, 225, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (223, 223+12, s)-Nets in Base 3

(223, 223+12, 6990613)-Net over F3 — Constructive and digital

(223, 223+12, large)-Net over F3 — Digital

(223, 223+12, large)-Net in Base 3 — Upper bound on s

(223, 223+12, 6990613)-Net over F₃ — Constructive and digital

(223, 223+12, large)-Net over F₃ — Digital