MinT - Best Known (199−18, 199, s)-Nets in Base 3

Best Known (199−18, 199, s)-Nets in Base 3

(199−18, 199, 932093)-Net over F₃ — Constructive and digital

Digital (181, 199, 932093)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (10, 19, 26)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 6, 14)-net over F₃, using
      - linear code embedding found in a computer search [i]
    2. digital (4, 13, 13)-net over F₃, using
      - a shift-net [i]
2. digital (162, 180, 932067)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸⁰, 932067, F₃, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(199−18, 199, 4194327)-Net over F₃ — Digital

Digital (181, 199, 4194327)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁹, 4194327, F₃, 2, 18) (dual of [(4194327, 2), 8388455, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁹, 26, F₃, 2, 9) (dual of [(26, 2), 33, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(3⁶, 14, F₃, 2, 4) (dual of [(14, 2), 22, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (2, 6, 14)-net over F₃, using
        linear code embedding found in a computer search [i]
      2. linear OOA(3¹³, 13, F₃, 2, 9) (dual of [(13, 2), 13, 10]-NRT-code), using
        extracting embedded OOA [i] based on digital (4, 13, 13)-net over F₃, using
        a shift-net [i]
  2. linear OOA(3¹⁸⁰, 4194301, F₃, 2, 18) (dual of [(4194301, 2), 8388422, 19]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁸⁰, 8388602, F₃, 18) (dual of [8388602, 8388422, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(199−18, 199, large)-Net in Base 3 — Upper bound on s

There is no (181, 199, large)-net in base 3, because

16 times m-reduction [i] would yield (181, 183, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]