MinT - Best Known (79−8, 79, s)-Nets in Base 3

Best Known (79−8, 79, s)-Nets in Base 3

(79−8, 79, 2097150)-Net over F₃ — Constructive and digital

Digital (71, 79, 2097150)-net over F₃, using

3³ times duplication [i] based on 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]

(79−8, 79, 4194302)-Net over F₃ — Digital

Digital (71, 79, 4194302)-net over F₃, using

3¹ times duplication [i] based on digital (70, 78, 4194302)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁷⁸, 4194302, F₃, 2, 8) (dual of [(4194302, 2), 8388526, 9]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(3⁷⁶, 4194301, F₃, 2, 8) (dual of [(4194301, 2), 8388526, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3⁷⁶, 8388602, F₃, 8) (dual of [8388602, 8388526, 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]

(79−8, 79, large)-Net in Base 3 — Upper bound on s

There is no (71, 79, large)-net in base 3, because

6 times m-reduction [i] would yield (71, 73, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]