MinT - Best Known (171−25, 171, s)-Nets in Base 3

Best Known (171−25, 171, s)-Nets in Base 3

(171−25, 171, 4924)-Net over F₃ — Constructive and digital

Digital (146, 171, 4924)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁷¹, 4924, F₃, 25, 25) (dual of [(4924, 25), 122929, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3¹⁷¹, 59089, F₃, 25) (dual of [59089, 58918, 26]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3¹⁶⁹, 59087, F₃, 25) (dual of [59087, 58918, 26]-code), using
    - constructionÂ X applied to C_e(24) âŠ‚ C_e(19) [i] based on
      1. linear OA(3¹⁶¹, 59049, F₃, 25) (dual of [59049, 58888, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      2. linear OA(3¹³¹, 59049, F₃, 20) (dual of [59049, 58918, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      3. linear OA(3⁸, 38, F₃, 4) (dual of [38, 30, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁸, 41, F₃, 4) (dual of [41, 33, 5]-code), using
        the narrow-sense BCH-code C(I) with length 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(171−25, 171, 20919)-Net over F₃ — Digital

Digital (146, 171, 20919)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷¹, 20919, F₃, 2, 25) (dual of [(20919, 2), 41667, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁷¹, 29544, F₃, 2, 25) (dual of [(29544, 2), 58917, 26]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁷⁰, 29544, F₃, 2, 25) (dual of [(29544, 2), 58918, 26]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁷⁰, 59088, F₃, 25) (dual of [59088, 58918, 26]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹⁶⁹, 59087, F₃, 25) (dual of [59087, 58918, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(19) [i] based on
        linear OA(3¹⁶¹, 59049, F₃, 25) (dual of [59049, 58888, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(3¹³¹, 59049, F₃, 20) (dual of [59049, 58918, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(3⁸, 38, F₃, 4) (dual of [38, 30, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁸, 41, F₃, 4) (dual of [41, 33, 5]-code), using
        the narrow-sense BCH-code C(I) with length 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(171−25, 171, large)-Net in Base 3 — Upper bound on s

There is no (146, 171, large)-net in base 3, because

23 times m-reduction [i] would yield (146, 148, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (171−25, 171, s)-Nets in Base 3

(171−25, 171, 4924)-Net over F3 — Constructive and digital

(171−25, 171, 20919)-Net over F3 — Digital

(171−25, 171, large)-Net in Base 3 — Upper bound on s

(171−25, 171, 4924)-Net over F₃ — Constructive and digital

(171−25, 171, 20919)-Net over F₃ — Digital