MinT - Best Known (66−6, 66, s)-Nets in Base 3

Best Known (66−6, 66, s)-Nets in Base 3

(66−6, 66, 2796257)-Net over F₃ — Constructive and digital

Digital (60, 66, 2796257)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁶⁶, 2796257, F₃, 6, 6) (dual of [(2796257, 6), 16777476, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁶⁶, 2796257, F₃, 5, 6) (dual of [(2796257, 5), 13981219, 7]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3⁶, 56, F₃, 5, 3) (dual of [(56, 5), 274, 4]-NRT-code), using
      - appending 2 arbitrary columns [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. linear OOA(3⁶⁰, 2796201, F₃, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(66−6, 66, large)-Net over F₃ — Digital

Digital (60, 66, large)-net over F₃, using

3⁶ times duplication [i] based on digital (54, 60, large)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3⁶⁰, large, F₃, 6, 6), using
  - appending kth column [i] based on linear OOA(3⁶⁰, large, F₃, 5, 6), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(66−6, 66, large)-Net in Base 3 — Upper bound on s

There is no (60, 66, large)-net in base 3, because

4 times m-reduction [i] would yield (60, 62, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]