MinT - Information on Result #662110

Information on Result #662110

Linear OA(4¹⁷, 68, F₄, 7) (dual of [68, 51, 8]-code), using generalized (u,Â u+v)-construction based on

linear OA(4¹, 17, F₄, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
  - parity-check code [i]
linear OA(4³, 17, F₄, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
  - Hamming code H(3,4) [i]
linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- ovoid in PG(3,Â 4) [i]
linear OA(4⁹, 17, F₄, 7) (dual of [17, 8, 8]-code), using
- dual code (with bound on d by constructionÂ Y1) [i] based on
  1. linear OA(4⁸, 17, F₄, 6) (dual of [17, 9, 7]-code), using
    - the BCH-code C(I) with length 17Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. nonexistence of linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), because
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, n and k, k and s, k and t, n and m, m and s, m and t, n and s, n and t.

Compare with Markus Grassl’s online database of code parameters.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Linear OA(4²³⁸, 4194373, F₄, 27) (dual of [4194373, 4194135, 28]-code)	[i]	ConstructionÂ X with Cyclic Codes
2	Linear OA(4²¹⁶, 4194373, F₄, 25) (dual of [4194373, 4194157, 26]-code)	[i]
3	Linear OA(4²⁵⁸, 1048645, F₄, 33) (dual of [1048645, 1048387, 34]-code)	[i]
4	Linear OA(4²¹⁸, 1048645, F₄, 27) (dual of [1048645, 1048427, 28]-code)	[i]
5	Linear OA(4¹⁹⁸, 1048645, F₄, 25) (dual of [1048645, 1048447, 26]-code)	[i]
6	Linear OA(4²⁵², 262213, F₄, 35) (dual of [262213, 261961, 36]-code)	[i]
7	Linear OA(4²³⁴, 262213, F₄, 33) (dual of [262213, 261979, 34]-code)	[i]
8	Linear OA(4¹⁹⁸, 262213, F₄, 27) (dual of [262213, 262015, 28]-code)	[i]
9	Linear OA(4¹⁸⁰, 262213, F₄, 25) (dual of [262213, 262033, 26]-code)	[i]
10	Linear OA(4²⁴⁸, 1048644, F₄, 31) (dual of [1048644, 1048396, 32]-code)	[i]	ConstructionÂ X with Extended Narrow-Sense BCH Codes
11	Linear OA(4²³⁸, 1048644, F₄, 30) (dual of [1048644, 1048406, 31]-code)	[i]
12	Linear OA(4²²⁸, 1048644, F₄, 29) (dual of [1048644, 1048416, 30]-code)	[i]
13	Linear OA(4²⁰⁸, 1048644, F₄, 26) (dual of [1048644, 1048436, 27]-code)	[i]
14	Linear OA(4²⁴³, 262212, F₄, 34) (dual of [262212, 261969, 35]-code)	[i]
15	Linear OA(4²²⁵, 262212, F₄, 31) (dual of [262212, 261987, 32]-code)	[i]
16	Linear OA(4²¹⁶, 262212, F₄, 30) (dual of [262212, 261996, 31]-code)	[i]
17	Linear OA(4²⁰⁷, 262212, F₄, 29) (dual of [262212, 262005, 30]-code)	[i]
18	Linear OA(4¹⁸⁹, 262212, F₄, 26) (dual of [262212, 262023, 27]-code)	[i]