MinT - Best Known (66−7, 66, s)-Nets in Base 3

Best Known (66−7, 66, s)-Nets in Base 3

(66−7, 66, 2796220)-Net over F₃ — Constructive and digital

Digital (59, 66, 2796220)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁶⁶, 2796220, F₃, 9, 7) (dual of [(2796220, 9), 25165914, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3⁶⁶, 2796221, F₃, 3, 7) (dual of [(2796221, 3), 8388597, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. 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. linear OOA(3⁶¹, 2796201, F₃, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(3⁶¹, large, F₃, 7) (dual of [large, large−61, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(66−7, 66, 4194321)-Net over F₃ — Digital

Digital (59, 66, 4194321)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁶⁶, 4194321, F₃, 2, 7) (dual of [(4194321, 2), 8388576, 8]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. 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. linear OOA(3⁶¹, 4194301, F₃, 2, 7) (dual of [(4194301, 2), 8388541, 8]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3⁶¹, 8388602, F₃, 7) (dual of [8388602, 8388541, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁶¹, large, F₃, 7) (dual of [large, large−61, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(66−7, 66, large)-Net in Base 3 — Upper bound on s

There is no (59, 66, large)-net in base 3, because

5 times m-reduction [i] would yield (59, 61, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]