MinT - Best Known (240−13, 240, s)-Nets in Base 3

Best Known (240−13, 240, s)-Nets in Base 3

(240−13, 240, 5788628)-Net over F₃ — Constructive and digital

Digital (227, 240, 5788628)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (51, 57, 1594328)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁵⁷, 1594328, F₃, 6, 6) (dual of [(1594328, 6), 9565911, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁵⁷, 1594328, F₃, 5, 6) (dual of [(1594328, 5), 7971583, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3⁵⁷, 4782984, F₃, 6) (dual of [4782984, 4782927, 7]-code), using
        1 times truncation [i] based on linear OA(3⁵⁸, 4782985, F₃, 7) (dual of [4782985, 4782927, 8]-code), using
        constructionÂ X4 applied to C_e(6) âŠ‚ C_e(4) [i] based on
        linear OA(3⁵⁷, 4782969, F₃, 7) (dual of [4782969, 4782912, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3⁴³, 4782969, F₃, 5) (dual of [4782969, 4782926, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3¹⁵, 16, F₃, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
        dual of repetition code with length 16 [i]
        linear OA(3¹, 16, F₃, 1) (dual of [16, 15, 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 (170, 183, 4194300)-net over F₃, using
  - trace code for nets [i] based on digital (48, 61, 1398100)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(240−13, 240, large)-Net over F₃ — Digital

Digital (227, 240, large)-net over F₃, using

t-expansion [i] based on digital (221, 240, large)-net over F₃, using
- 1 times m-reduction [i] based on digital (221, 241, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, large, F₃, 20) (dual of [large, large−241, 21]-code), using
    - strength reduction [i] based on linear OA(3²⁴¹, large, F₃, 25) (dual of [large, large−241, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(240−13, 240, large)-Net in Base 3 — Upper bound on s

There is no (227, 240, large)-net in base 3, because

11 times m-reduction [i] would yield (227, 229, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (240−13, 240, s)-Nets in Base 3

(240−13, 240, 5788628)-Net over F3 — Constructive and digital

(240−13, 240, large)-Net over F3 — Digital

(240−13, 240, large)-Net in Base 3 — Upper bound on s

(240−13, 240, 5788628)-Net over F₃ — Constructive and digital

(240−13, 240, large)-Net over F₃ — Digital