MinT - Best Known (137, 150, s)-Nets in Base 3

Best Known (137, 150, s)-Nets in Base 3

(137, 150, 1398831)-Net over F₃ — Constructive and digital

Digital (137, 150, 1398831)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (23, 29, 731)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁹, 731, F₃, 6, 6) (dual of [(731, 6), 4357, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3²⁹, 731, F₃, 5, 6) (dual of [(731, 5), 3626, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3²⁹, 2193, F₃, 6) (dual of [2193, 2164, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3²⁹, 2195, F₃, 6) (dual of [2195, 2166, 7]-code), using
        1 times truncation [i] based on linear OA(3³⁰, 2196, F₃, 7) (dual of [2196, 2166, 8]-code), using
        constructionÂ X4 applied to C_e(6) âŠ‚ C_e(4) [i] based on
        linear OA(3²⁹, 2187, F₃, 7) (dual of [2187, 2158, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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(3⁸, 9, F₃, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
        dual of repetition code with length 9 [i]
        linear OA(3¹, 9, F₃, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (108, 121, 1398100)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹²¹, 1398100, F₃, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(3¹²¹, 8388601, F₃, 13) (dual of [8388601, 8388480, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹²¹, large, F₃, 13) (dual of [large, large−121, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(137, 150, 7125377)-Net over F₃ — Digital

Digital (137, 150, 7125377)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁰, 7125377, F₃, 13) (dual of [7125377, 7125227, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁰, large, F₃, 13) (dual of [large, large−150, 14]-code), using
  - strength reduction [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]

(137, 150, large)-Net in Base 3 — Upper bound on s

There is no (137, 150, large)-net in base 3, because

11 times m-reduction [i] would yield (137, 139, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (137, 150, s)-Nets in Base 3

(137, 150, 1398831)-Net over F3 — Constructive and digital

(137, 150, 7125377)-Net over F3 — Digital

(137, 150, large)-Net in Base 3 — Upper bound on s

(137, 150, 1398831)-Net over F₃ — Constructive and digital

(137, 150, 7125377)-Net over F₃ — Digital