MinT - Best Known (161−15, 161, s)-Nets in Base 3

Best Known (161−15, 161, s)-Nets in Base 3

(161−15, 161, 1198386)-Net over F₃ — Constructive and digital

Digital (146, 161, 1198386)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 15)-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]

(161−15, 161, 4194316)-Net over F₃ — Digital

Digital (146, 161, 4194316)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶¹, 4194316, F₃, 2, 15) (dual of [(4194316, 2), 8388471, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹¹, 15, F₃, 2, 7) (dual of [(15, 2), 19, 8]-NRT-code), using
    - extracting embedded OOA [i] based on digital (4, 11, 15)-net over F₃, using
      - linear code embedding found in a computer search [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]

(161−15, 161, large)-Net in Base 3 — Upper bound on s

There is no (146, 161, large)-net in base 3, because

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