MinT - Best Known (81, 81+8, s)-Nets in Base 3

Best Known (81, 81+8, s)-Nets in Base 3

(81, 81+8, 2097878)-Net over F₃ — Constructive and digital

Digital (81, 89, 2097878)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 13, 728)-net over F₃, using
  - nets constructed from net-embeddable BCH codes [i]
2. digital (68, 76, 2097150)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁷⁶, 2097150, F₃, 8, 8) (dual of [(2097150, 8), 16777124, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(3⁷⁶, 8388600, F₃, 8) (dual of [8388600, 8388524, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷⁶, large, F₃, 8) (dual of [large, large−76, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(81, 81+8, large)-Net over F₃ — Digital

Digital (81, 89, large)-net over F₃, using

3³ times duplication [i] based on digital (78, 86, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸⁶, large, F₃, 8) (dual of [large, large−86, 9]-code), using
  - 10 times code embedding in larger space [i] based on linear OA(3⁷⁶, large, F₃, 8) (dual of [large, large−76, 9]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(81, 81+8, large)-Net in Base 3 — Upper bound on s

There is no (81, 89, large)-net in base 3, because

6 times m-reduction [i] would yield (81, 83, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]