MinT - Best Known (142, 142+14, s)-Nets in Base 3

Best Known (142, 142+14, s)-Nets in Base 3

(142, 142+14, 1198439)-Net over F₃ — Constructive and digital

Digital (142, 156, 1198439)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 20, 68)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 6, 56)-net over F₃, using
      - net defined by OOA [i] based on linear OOA(3⁶, 56, F₃, 3, 3) (dual of [(56, 3), 162, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(3⁶, 56, F₃, 2, 3) (dual of [(56, 2), 106, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [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]
    2. digital (7, 14, 34)-net over F₃, using
      - linear code embedding found in a computer search [i]
2. digital (122, 136, 1198371)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³⁶, 1198371, F₃, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(3¹³⁶, 8388597, F₃, 14) (dual of [8388597, 8388461, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(142, 142+14, 4194382)-Net over F₃ — Digital

Digital (142, 156, 4194382)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁶, 4194382, F₃, 2, 14) (dual of [(4194382, 2), 8388608, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁰, 81, F₃, 2, 7) (dual of [(81, 2), 142, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰, 81, F₃, 7) (dual of [81, 61, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰, 121, F₃, 7) (dual of [121, 101, 8]-code), using
        the BCH-code C(I) with length 121Â | 3⁵âˆ’1, defining interval IÂ =Â {−3,−1,1,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
  2. linear OOA(3¹³⁶, 4194301, F₃, 2, 14) (dual of [(4194301, 2), 8388466, 15]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹³⁶, 8388602, F₃, 14) (dual of [8388602, 8388466, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(142, 142+14, large)-Net in Base 3 — Upper bound on s

There is no (142, 156, large)-net in base 3, because

12 times m-reduction [i] would yield (142, 144, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]