MinT - Best Known (229−22, 229, s)-Nets in Base 3

Best Known (229−22, 229, s)-Nets in Base 3

(229−22, 229, 762618)-Net over F₃ — Constructive and digital

Digital (207, 229, 762618)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 18)-net over F₃, using
  - a shift-net [i]
2. digital (189, 211, 762600)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹¹, 762600, F₃, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3²¹¹, 8388600, F₃, 22) (dual of [8388600, 8388389, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(229−22, 229, 2796219)-Net over F₃ — Digital

Digital (207, 229, 2796219)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁹, 2796219, F₃, 3, 22) (dual of [(2796219, 3), 8388428, 23]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁸, 18, F₃, 3, 11) (dual of [(18, 3), 36, 12]-NRT-code), using
    - extracting embedded OOA [i] based on digital (7, 18, 18)-net over F₃, using
      - a shift-net [i]
  2. linear OOA(3²¹¹, 2796201, F₃, 3, 22) (dual of [(2796201, 3), 8388392, 23]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(229−22, 229, large)-Net in Base 3 — Upper bound on s

There is no (207, 229, large)-net in base 3, because

20 times m-reduction [i] would yield (207, 209, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]