MinT - Best Known (89, 89+11, s)-Nets in Base 7

Best Known (89, 89+11, s)-Nets in Base 7

(89, 89+11, 2308672)-Net over F₇ — Constructive and digital

Digital (89, 100, 2308672)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (13, 18, 2752)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7¹⁸, 2752, F₇, 6, 5) (dual of [(2752, 6), 16494, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(7¹⁸, 2753, F₇, 2, 5) (dual of [(2753, 2), 5488, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(7⁴, 400, F₇, 2, 2) (dual of [(400, 2), 796, 3]-NRT-code), using
        appending kth column [i] based on linear OA(7⁴, 400, F₇, 2) (dual of [400, 396, 3]-code), using
        Hamming code H(4,7) [i]
        linear OOA(7¹⁴, 2353, F₇, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(7¹⁴, 4706, F₇, 5) (dual of [4706, 4692, 6]-code), using
        trace code [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (71, 82, 2305920)-net over F₇, using
  - trace code for nets [i] based on digital (30, 41, 1152960)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁴¹, 1152960, F₄₉, 11, 11) (dual of [(1152960, 11), 12682519, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(49⁴¹, 5764801, F₄₉, 11) (dual of [5764801, 5764760, 12]-code), using
        an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(89, 89+11, large)-Net over F₇ — Digital

Digital (89, 100, large)-net over F₇, using

7² times duplication [i] based on digital (87, 98, large)-net over F₇, using
- t-expansion [i] based on digital (85, 98, large)-net over F₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁸, large, F₇, 13) (dual of [large, large−98, 14]-code), using
    - trace code [i] based on linear OA(49⁴⁹, 5764802, F₄₉, 13) (dual of [5764802, 5764753, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 5764802Â | 49⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(89, 89+11, large)-Net in Base 7 — Upper bound on s

There is no (89, 100, large)-net in base 7, because

9 times m-reduction [i] would yield (89, 91, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 89+11, s)-Nets in Base 7

(89, 89+11, 2308672)-Net over F7 — Constructive and digital

(89, 89+11, large)-Net over F7 — Digital

(89, 89+11, large)-Net in Base 7 — Upper bound on s

(89, 89+11, 2308672)-Net over F₇ — Constructive and digital

(89, 89+11, large)-Net over F₇ — Digital