MinT - Best Known (172−13, 172, s)-Nets in Base 3

Best Known (172−13, 172, s)-Nets in Base 3

(172−13, 172, 1594371)-Net over F₃ — Constructive and digital

Digital (159, 172, 1594371)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (10, 16, 46)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶, 46, F₃, 6, 6) (dual of [(46, 6), 260, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁶, 46, F₃, 5, 6) (dual of [(46, 5), 214, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3³, 4, F₃, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(5;17,3) [i]
        linear OOA(3¹³, 42, F₃, 5, 6) (dual of [(42, 5), 197, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 13, 42)-net over F₃, using
        linear code embedding found in a computer search [i]
2. digital (143, 156, 1594325)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵⁶, 1594325, F₃, 14, 13) (dual of [(1594325, 14), 22320394, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(3¹⁵⁶, 4782976, F₃, 2, 13) (dual of [(4782976, 2), 9565796, 14]-NRT-code), using
      - trace code [i] based on linear OOA(9⁷⁸, 2391488, F₉, 2, 13) (dual of [(2391488, 2), 4782898, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(9⁷⁸, 4782976, F₉, 13) (dual of [4782976, 4782898, 14]-code), using
        constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(9⁷⁸, 4782969, F₉, 13) (dual of [4782969, 4782891, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(9⁷¹, 4782969, F₉, 12) (dual of [4782969, 4782898, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(9⁰, 7, F₉, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(172−13, 172, large)-Net over F₃ — Digital

Digital (159, 172, large)-net over F₃, using

3⁷ times duplication [i] based on digital (152, 165, large)-net over F₃, using
- t-expansion [i] based on digital (151, 165, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁵, large, F₃, 14) (dual of [large, large−165, 15]-code), using
    - 29 times code embedding in larger space [i] based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(172−13, 172, large)-Net in Base 3 — Upper bound on s

There is no (159, 172, large)-net in base 3, because

11 times m-reduction [i] would yield (159, 161, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (172−13, 172, s)-Nets in Base 3

(172−13, 172, 1594371)-Net over F3 — Constructive and digital

(172−13, 172, large)-Net over F3 — Digital

(172−13, 172, large)-Net in Base 3 — Upper bound on s

(172−13, 172, 1594371)-Net over F₃ — Constructive and digital

(172−13, 172, large)-Net over F₃ — Digital