MinT - Information on Result #894903

Information on Result #894903

Linear OOA(3⁶⁸, 4194413, F₃, 2, 7) (dual of [(4194413, 2), 8388758, 8]-NRT-code), using (u,Â u+v)-construction based on

linear OOA(3⁷, 112, F₃, 2, 3) (dual of [(112, 2), 217, 4]-NRT-code), using
- OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁷, 112, F₃, 3) (dual of [112, 105, 4]-code or 112-cap in PG(6,3)), using
  - doubling the cap [i] based on linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
    - the Hill cap [i]
linear OOA(3⁶¹, 4194301, F₃, 2, 7) (dual of [(4194301, 2), 8388541, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁶¹, 8388602, F₃, 7) (dual of [8388602, 8388541, 8]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁶¹, large, F₃, 7) (dual of [large, large−61, 8]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

Mode: Constructive and linear.

None.

The following results depend on this result:

	Result		Method
1	Linear OOA(3⁶⁸, 4194413, F₃, 3, 7) (dual of [(4194413, 3), 12583171, 8]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(3⁶⁸, 4194413, F₃, 4, 7) (dual of [(4194413, 4), 16777584, 8]-NRT-code)	[i]
3	Linear OOA(3⁶⁸, 4194413, F₃, 5, 7) (dual of [(4194413, 5), 20971997, 8]-NRT-code)	[i]
4	Digital (61, 68, 4194413)-net over F₃	[i]