MinT - Best Known (180−11, 180, s)-Nets in Base 3

Best Known (180−11, 180, s)-Nets in Base 3

(180−11, 180, 6711037)-Net over F₃ — Constructive and digital

Digital (169, 180, 6711037)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 16, 157)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶, 157, F₃, 5, 5) (dual of [(157, 5), 769, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁶, 157, F₃, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 40, F₃, 4, 1) (dual of [(40, 4), 159, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(3¹, s, F₃, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(3⁴, 40, F₃, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁴, 40, F₃, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]
        linear OOA(3¹¹, 77, F₃, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 11, 77)-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]

(180−11, 180, large)-Net over F₃ — Digital

Digital (169, 180, large)-net over F₃, using

3² times duplication [i] based on 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]

(180−11, 180, large)-Net in Base 3 — Upper bound on s

There is no (169, 180, large)-net in base 3, because

9 times m-reduction [i] would yield (169, 171, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (180−11, 180, s)-Nets in Base 3

(180−11, 180, 6711037)-Net over F3 — Constructive and digital

(180−11, 180, large)-Net over F3 — Digital

(180−11, 180, large)-Net in Base 3 — Upper bound on s

(180−11, 180, 6711037)-Net over F₃ — Constructive and digital

(180−11, 180, large)-Net over F₃ — Digital