MinT - Best Known (152−19, 152, s)-Nets in Base 3

Best Known (152−19, 152, s)-Nets in Base 3

(152−19, 152, 59052)-Net over F₃ — Constructive and digital

Digital (133, 152, 59052)-net over F₃, using

3³ times duplication [i] based on digital (130, 149, 59052)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁴⁹, 59052, F₃, 19, 19) (dual of [(59052, 19), 1121839, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(3¹⁴⁹, 531469, F₃, 19) (dual of [531469, 531320, 20]-code), using
    - constructionÂ X applied to C_e(18) âŠ‚ C_e(15) [i] based on
      1. linear OA(3¹⁴⁵, 531441, F₃, 19) (dual of [531441, 531296, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      2. linear OA(3¹²¹, 531441, F₃, 16) (dual of [531441, 531320, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(3⁴, 28, F₃, 2) (dual of [28, 24, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(152−19, 152, 176241)-Net over F₃ — Digital

Digital (133, 152, 176241)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵², 176241, F₃, 3, 19) (dual of [(176241, 3), 528571, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵², 177157, F₃, 3, 19) (dual of [(177157, 3), 531319, 20]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁵¹, 177157, F₃, 3, 19) (dual of [(177157, 3), 531320, 20]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁵¹, 531471, F₃, 19) (dual of [531471, 531320, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵¹, 531472, F₃, 19) (dual of [531472, 531321, 20]-code), using
        constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
        linear OA(3¹⁴⁵, 531442, F₃, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3¹²¹, 531442, F₃, 15) (dual of [531442, 531321, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(3⁶, 30, F₃, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(152−19, 152, large)-Net in Base 3 — Upper bound on s

There is no (133, 152, large)-net in base 3, because

17 times m-reduction [i] would yield (133, 135, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (152−19, 152, s)-Nets in Base 3

(152−19, 152, 59052)-Net over F3 — Constructive and digital

(152−19, 152, 176241)-Net over F3 — Digital

(152−19, 152, large)-Net in Base 3 — Upper bound on s

(152−19, 152, 59052)-Net over F₃ — Constructive and digital

(152−19, 152, 176241)-Net over F₃ — Digital