MinT - Best Known (166−15, 166, s)-Nets in Base 3

Best Known (166−15, 166, s)-Nets in Base 3

(166−15, 166, 1198405)-Net over F₃ — Constructive and digital

Digital (151, 166, 1198405)-net over F₃, using

3² times duplication [i] based on digital (149, 164, 1198405)-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 (135, 150, 1198371)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3¹⁵⁰, 1198371, F₃, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(3¹⁵⁰, 8388598, F₃, 15) (dual of [8388598, 8388448, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵⁰, large, F₃, 15) (dual of [large, large−150, 16]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(166−15, 166, 4194342)-Net over F₃ — Digital

Digital (151, 166, 4194342)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁶, 4194342, F₃, 2, 15) (dual of [(4194342, 2), 8388518, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁶, 41, F₃, 2, 7) (dual of [(41, 2), 66, 8]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
      - generator matrices provided on Edelâ€™s homepage [i]
  2. linear OOA(3¹⁵⁰, 4194301, F₃, 2, 15) (dual of [(4194301, 2), 8388452, 16]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁵⁰, 8388602, F₃, 15) (dual of [8388602, 8388452, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵⁰, large, F₃, 15) (dual of [large, large−150, 16]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(166−15, 166, large)-Net in Base 3 — Upper bound on s

There is no (151, 166, large)-net in base 3, because

13 times m-reduction [i] would yield (151, 153, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]