MinT - Best Known (166−14, 166, s)-Nets in Base 8

Best Known (166−14, 166, s)-Nets in Base 8

(166−14, 166, 4795500)-Net over F₈ — Constructive and digital

Digital (152, 166, 4795500)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 11, 2016)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹¹, 2016, F₈, 4, 4) (dual of [(2016, 4), 8053, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹¹, 2016, F₈, 3, 4) (dual of [(2016, 3), 6037, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹¹, 4032, F₈, 4) (dual of [4032, 4021, 5]-code), using
        base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [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]

(166−14, 166, large)-Net over F₈ — Digital

Digital (152, 166, large)-net over F₈, using

t-expansion [i] based on digital (144, 166, large)-net over F₈, using
- 2 times m-reduction [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]

(166−14, 166, large)-Net in Base 8 — Upper bound on s

There is no (152, 166, large)-net in base 8, because

12 times m-reduction [i] would yield (152, 154, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (166−14, 166, s)-Nets in Base 8

(166−14, 166, 4795500)-Net over F8 — Constructive and digital

(166−14, 166, large)-Net over F8 — Digital

(166−14, 166, large)-Net in Base 8 — Upper bound on s

(166−14, 166, 4795500)-Net over F₈ — Constructive and digital

(166−14, 166, large)-Net over F₈ — Digital