MinT - Best Known (191−25, 191, s)-Nets in Base 4

Best Known (191−25, 191, s)-Nets in Base 4

(191−25, 191, 87385)-Net over F₄ — Constructive and digital

Digital (166, 191, 87385)-net over F₄, using

net defined by OOA [i] based on linear OOA(4¹⁹¹, 87385, F₄, 25, 25) (dual of [(87385, 25), 2184434, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4¹⁹¹, 1048621, F₄, 25) (dual of [1048621, 1048430, 26]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹⁹¹, 1048627, F₄, 25) (dual of [1048627, 1048436, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(4¹⁸¹, 1048577, F₄, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(4¹⁴¹, 1048577, F₄, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(4¹⁰, 50, F₄, 5) (dual of [50, 40, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(191−25, 191, 448702)-Net over F₄ — Digital

Digital (166, 191, 448702)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁹¹, 448702, F₄, 2, 25) (dual of [(448702, 2), 897213, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4¹⁹¹, 524313, F₄, 2, 25) (dual of [(524313, 2), 1048435, 26]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4¹⁹¹, 1048626, F₄, 25) (dual of [1048626, 1048435, 26]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹⁹¹, 1048627, F₄, 25) (dual of [1048627, 1048436, 26]-code), using
      - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
        linear OA(4¹⁸¹, 1048577, F₄, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(4¹⁴¹, 1048577, F₄, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(4¹⁰, 50, F₄, 5) (dual of [50, 40, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(191−25, 191, large)-Net in Base 4 — Upper bound on s

There is no (166, 191, large)-net in base 4, because

23 times m-reduction [i] would yield (166, 168, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (191−25, 191, s)-Nets in Base 4

(191−25, 191, 87385)-Net over F4 — Constructive and digital

(191−25, 191, 448702)-Net over F4 — Digital

(191−25, 191, large)-Net in Base 4 — Upper bound on s

(191−25, 191, 87385)-Net over F₄ — Constructive and digital

(191−25, 191, 448702)-Net over F₄ — Digital