MinT - Best Known (146, 146+16, s)-Nets in Base 3

Best Known (146, 146+16, s)-Nets in Base 3

(146, 146+16, 1048586)-Net over F₃ — Constructive and digital

Digital (146, 162, 1048586)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 11)-net over F₃, using
  - a shift-net [i]
2. digital (135, 151, 1048575)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵¹, 1048575, F₃, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(3¹⁵¹, 8388600, F₃, 16) (dual of [8388600, 8388449, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(146, 146+16, 2796212)-Net over F₃ — Digital

Digital (146, 162, 2796212)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶², 2796212, F₃, 3, 16) (dual of [(2796212, 3), 8388474, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹¹, 11, F₃, 3, 8) (dual of [(11, 3), 22, 9]-NRT-code), using
    - extracting embedded OOA [i] based on digital (3, 11, 11)-net over F₃, using
      - a shift-net [i]
  2. linear OOA(3¹⁵¹, 2796201, F₃, 3, 16) (dual of [(2796201, 3), 8388452, 17]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(146, 146+16, large)-Net in Base 3 — Upper bound on s

There is no (146, 162, large)-net in base 3, because

14 times m-reduction [i] would yield (146, 148, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]