MinT - Information on Result #2257416

Information on Result #2257416

Linear OA(2⁷⁸, 32786, F₂, 11) (dual of [32786, 32708, 12]-code), using 1 times code embedding in larger space based on linear OA(2⁷⁷, 32785, F₂, 11) (dual of [32785, 32708, 12]-code), using

constructionÂ X4 applied to C_e(10) âŠ‚ C_e(8) [i] based on
1. linear OA(2⁷⁶, 32768, F₂, 11) (dual of [32768, 32692, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 2¹⁵âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
2. linear OA(2⁶¹, 32768, F₂, 9) (dual of [32768, 32707, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 2¹⁵âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
3. linear OA(2¹⁶, 17, F₂, 15) (dual of [17, 1, 16]-code), using
  - strength reduction [i] based on linear OA(2¹⁶, 17, F₂, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
    - dual of repetition code with length 17 [i]
4. linear OA(2¹, 17, F₂, 1) (dual of [17, 16, 2]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
    - parity-check code [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.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Linear OOA(2⁷⁸, 16393, F₂, 2, 11) (dual of [(16393, 2), 32708, 12]-NRT-code)	[i]	OOA Folding
2	Linear OOA(2⁷⁸, 6557, F₂, 6, 11) (dual of [(6557, 6), 39264, 12]-NRT-code)	[i]	OOA Folding and Stacking with Additional Row
3	Linear OOA(2⁷⁸, 6557, F₂, 7, 11) (dual of [(6557, 7), 45821, 12]-NRT-code)	[i]
4	Linear OOA(2⁷⁸, 6557, F₂, 8, 11) (dual of [(6557, 8), 52378, 12]-NRT-code)	[i]
5	Linear OOA(2⁷⁸, 6557, F₂, 11, 11) (dual of [(6557, 11), 72049, 12]-NRT-code)	[i]