MinT - Best Known (240−12, 240, s)-Nets in Base 3

Best Known (240−12, 240, s)-Nets in Base 3

(240−12, 240, 8388601)-Net over F₃ — Constructive and digital

Digital (228, 240, 8388601)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (54, 60, 2796201)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁶⁰, 2796201, F₃, 6, 6) (dual of [(2796201, 6), 16777146, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁶⁰, 2796201, F₃, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
2. digital (168, 180, 5592400)-net over F₃, using
  - trace code for nets [i] based on digital (78, 90, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁰, 2796200, F₉, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁰, 8388601, F₉, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁰, 8388602, F₉, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁵, 4194301, F₈₁, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁵, 8388602, F₈₁, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(240−12, 240, large)-Net over F₃ — Digital

Digital (228, 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]

(240−12, 240, large)-Net in Base 3 — Upper bound on s

There is no (228, 240, large)-net in base 3, because

10 times m-reduction [i] would yield (228, 230, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (240−12, 240, s)-Nets in Base 3

(240−12, 240, 8388601)-Net over F3 — Constructive and digital

(240−12, 240, large)-Net over F3 — Digital

(240−12, 240, large)-Net in Base 3 — Upper bound on s

(240−12, 240, 8388601)-Net over F₃ — Constructive and digital

(240−12, 240, large)-Net over F₃ — Digital