MinT - Best Known (244−23, 244, s)-Nets in Base 3

Best Known (244−23, 244, s)-Nets in Base 3

(244−23, 244, 762618)-Net over F₃ — Constructive and digital

Digital (221, 244, 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 (203, 226, 762600)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²²⁶, 762600, F₃, 23, 23) (dual of [(762600, 23), 17539574, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(3²²⁶, 8388601, F₃, 23) (dual of [8388601, 8388375, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²²⁶, large, F₃, 23) (dual of [large, large−226, 24]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(244−23, 244, 2796219)-Net over F₃ — Digital

Digital (221, 244, 2796219)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁴, 2796219, F₃, 3, 23) (dual of [(2796219, 3), 8388413, 24]-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(3²²⁶, 2796201, F₃, 3, 23) (dual of [(2796201, 3), 8388377, 24]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3²²⁶, large, F₃, 23) (dual of [large, large−226, 24]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(244−23, 244, large)-Net in Base 3 — Upper bound on s

There is no (221, 244, large)-net in base 3, because

21 times m-reduction [i] would yield (221, 223, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]