MinT - Best Known (191−28, 191, s)-Nets in Base 3

Best Known (191−28, 191, s)-Nets in Base 3

(191−28, 191, 4220)-Net over F₃ — Constructive and digital

Digital (163, 191, 4220)-net over F₃, using

3² times duplication [i] based on digital (161, 189, 4220)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁸⁹, 4220, F₃, 28, 28) (dual of [(4220, 28), 117971, 29]-NRT-code), using
  - OA 14-folding and stacking [i] based on linear OA(3¹⁸⁹, 59080, F₃, 28) (dual of [59080, 58891, 29]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁸⁹, 59087, F₃, 28) (dual of [59087, 58898, 29]-code), using
      - constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
        linear OA(3¹⁸¹, 59049, F₃, 28) (dual of [59049, 58868, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [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]

(191−28, 191, 20565)-Net over F₃ — Digital

Digital (163, 191, 20565)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹¹, 20565, F₃, 2, 28) (dual of [(20565, 2), 40939, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁹¹, 29544, F₃, 2, 28) (dual of [(29544, 2), 58897, 29]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁹⁰, 29544, F₃, 2, 28) (dual of [(29544, 2), 58898, 29]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁹⁰, 59088, F₃, 28) (dual of [59088, 58898, 29]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹⁸⁹, 59087, F₃, 28) (dual of [59087, 58898, 29]-code), using
        constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
        linear OA(3¹⁸¹, 59049, F₃, 28) (dual of [59049, 58868, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [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]

(191−28, 191, large)-Net in Base 3 — Upper bound on s

There is no (163, 191, large)-net in base 3, because

26 times m-reduction [i] would yield (163, 165, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (191−28, 191, s)-Nets in Base 3

(191−28, 191, 4220)-Net over F3 — Constructive and digital

(191−28, 191, 20565)-Net over F3 — Digital

(191−28, 191, large)-Net in Base 3 — Upper bound on s

(191−28, 191, 4220)-Net over F₃ — Constructive and digital

(191−28, 191, 20565)-Net over F₃ — Digital