MinT - Best Known (188−11, 188, s)-Nets in Base 3

Best Known (188−11, 188, s)-Nets in Base 3

(188−11, 188, 6711980)-Net over F₃ — Constructive and digital

Digital (177, 188, 6711980)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (19, 24, 1100)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁴, 1100, F₃, 5, 5) (dual of [(1100, 5), 5476, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(3²⁴, 1101, F₃, 2, 5) (dual of [(1101, 2), 2178, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3², 4, F₃, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;6,3) [i]
        linear OOA(3²², 1097, F₃, 2, 5) (dual of [(1097, 2), 2172, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(3²², 2194, F₃, 5) (dual of [2194, 2172, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(3²², 2187, F₃, 5) (dual of [2187, 2165, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3¹⁵, 2187, F₃, 4) (dual of [2187, 2172, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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]

(188−11, 188, large)-Net over F₃ — Digital

Digital (177, 188, large)-net over F₃, using

t-expansion [i] based on digital (175, 188, large)-net over F₃, using
- 3 times m-reduction [i] based on digital (175, 191, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹¹, large, F₃, 16) (dual of [large, large−191, 17]-code), using
    - 40 times code embedding in larger space [i] based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(188−11, 188, large)-Net in Base 3 — Upper bound on s

There is no (177, 188, large)-net in base 3, because

9 times m-reduction [i] would yield (177, 179, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (188−11, 188, s)-Nets in Base 3

(188−11, 188, 6711980)-Net over F3 — Constructive and digital

(188−11, 188, large)-Net over F3 — Digital

(188−11, 188, large)-Net in Base 3 — Upper bound on s

(188−11, 188, 6711980)-Net over F₃ — Constructive and digital

(188−11, 188, large)-Net over F₃ — Digital