MinT - Best Known (178−11, 178, s)-Nets in Base 3

Best Known (178−11, 178, s)-Nets in Base 3

(178−11, 178, 6710983)-Net over F₃ — Constructive and digital

Digital (167, 178, 6710983)-net over F₃, using

3¹ times duplication [i] based on digital (166, 177, 6710983)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 13, 103)-net over F₃, using
    - linear code embedding found in a computer search [i]
  2. digital (153, 164, 6710880)-net over F₃, using
    - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁴¹, 1677720, F₈₁, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(81⁴¹, 8388601, F₈₁, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴¹, large, F₈₁, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(178−11, 178, large)-Net over F₃ — Digital

Digital (167, 178, large)-net over F₃, using

t-expansion [i] based on digital (163, 178, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁸, large, F₃, 15) (dual of [large, large−178, 16]-code), using
  - 28 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]

(178−11, 178, large)-Net in Base 3 — Upper bound on s

There is no (167, 178, large)-net in base 3, because

9 times m-reduction [i] would yield (167, 169, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]