MinT - Best Known (170−15, 170, s)-Nets in Base 3

Best Known (170−15, 170, s)-Nets in Base 3

(170−15, 170, 1198439)-Net over F₃ — Constructive and digital

Digital (155, 170, 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 (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]

(170−15, 170, 4518053)-Net over F₃ — Digital

Digital (155, 170, 4518053)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁰, 4518053, F₃, 15) (dual of [4518053, 4517883, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁷⁰, large, F₃, 15) (dual of [large, large−170, 16]-code), using
  - 20 times code embedding in larger space [i] 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]

(170−15, 170, large)-Net in Base 3 — Upper bound on s

There is no (155, 170, large)-net in base 3, because

13 times m-reduction [i] would yield (155, 157, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]