MinT - Best Known (168−14, 168, s)-Nets in Base 8

Best Known (168−14, 168, s)-Nets in Base 8

(168−14, 168, 4797516)-Net over F₈ — Constructive and digital

Digital (154, 168, 4797516)-net over F₈, using

8¹ times duplication [i] based on digital (153, 167, 4797516)-net over F₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (8, 12, 4032)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8¹², 4032, F₈, 4, 4) (dual of [(4032, 4), 16116, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(8¹², 4032, F₈, 3, 4) (dual of [(4032, 3), 12084, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(8¹², 8064, F₈, 4) (dual of [8064, 8052, 5]-code), using
        trace code [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
        1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (42, 49, 2396742)-net over F₈, using
    - s-reduction based on digital (42, 49, 2796200)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8⁴⁹, 2796200, F₈, 7, 7) (dual of [(2796200, 7), 19573351, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(8⁴⁹, 8388601, F₈, 7) (dual of [8388601, 8388552, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁴⁹, large, F₈, 7) (dual of [large, large−49, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
  3. digital (92, 106, 2396742)-net over F₈, using
    - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(168−14, 168, large)-Net over F₈ — Digital

Digital (154, 168, large)-net over F₈, using

t-expansion [i] based on digital (144, 168, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(168−14, 168, large)-Net in Base 8 — Upper bound on s

There is no (154, 168, large)-net in base 8, because

12 times m-reduction [i] would yield (154, 156, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (168−14, 168, s)-Nets in Base 8

(168−14, 168, 4797516)-Net over F8 — Constructive and digital

(168−14, 168, large)-Net over F8 — Digital

(168−14, 168, large)-Net in Base 8 — Upper bound on s

(168−14, 168, 4797516)-Net over F₈ — Constructive and digital

(168−14, 168, large)-Net over F₈ — Digital