MinT - Best Known (168−15, 168, s)-Nets in Base 3

Best Known (168−15, 168, s)-Nets in Base 3

(168−15, 168, 1198417)-Net over F₃ — Constructive and digital

Digital (153, 168, 1198417)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 18, 46)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 20)-net over F₃, using
      - net defined by OOA [i] based on linear OOA(3⁵, 20, F₃, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(3⁵, 20, F₃, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
        doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]
    2. digital (6, 13, 26)-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]

(168−15, 168, 4194352)-Net over F₃ — Digital

Digital (153, 168, 4194352)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁸, 4194352, F₃, 2, 15) (dual of [(4194352, 2), 8388536, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁸, 51, F₃, 2, 7) (dual of [(51, 2), 84, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸, 51, F₃, 7) (dual of [51, 33, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸, 86, F₃, 7) (dual of [86, 68, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(4) [i] based on
        linear OA(3¹⁷, 81, F₃, 7) (dual of [81, 64, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3¹³, 81, F₃, 5) (dual of [81, 68, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3¹, 5, F₃, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [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]

(168−15, 168, large)-Net in Base 3 — Upper bound on s

There is no (153, 168, large)-net in base 3, because

13 times m-reduction [i] would yield (153, 155, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (168−15, 168, s)-Nets in Base 3

(168−15, 168, 1198417)-Net over F3 — Constructive and digital

(168−15, 168, 4194352)-Net over F3 — Digital

(168−15, 168, large)-Net in Base 3 — Upper bound on s

(168−15, 168, 1198417)-Net over F₃ — Constructive and digital

(168−15, 168, 4194352)-Net over F₃ — Digital