MinT - Information on Result #2295450

Information on Result #2295450

Linear OA(2⁸⁴, 8212, F₂, 13) (dual of [8212, 8128, 14]-code), using adding a parity check bit based on linear OA(2⁸³, 8211, F₂, 12) (dual of [8211, 8128, 13]-code), using

constructionÂ X with VarÅ¡amov bound [i] based on
1. linear OA(2⁷⁹, 8206, F₂, 12) (dual of [8206, 8127, 13]-code), using
  - 1 times truncation [i] based on linear OA(2⁸⁰, 8207, F₂, 13) (dual of [8207, 8127, 14]-code), using
    - constructionÂ X4 applied to C_e(12) âŠ‚ C_e(10) [i] based on
      1. linear OA(2⁷⁹, 8192, F₂, 13) (dual of [8192, 8113, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(2⁶⁶, 8192, F₂, 11) (dual of [8192, 8126, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(2¹⁴, 15, F₂, 13) (dual of [15, 1, 14]-code), using
        strength reduction [i] based on linear OA(2¹⁴, 15, F₂, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
        dual of repetition code with length 15 [i]
      4. linear OA(2¹, 15, F₂, 1) (dual of [15, 14, 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]
2. linear OA(2⁷⁹, 8207, F₂, 8) (dual of [8207, 8128, 9]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 2⁷⁹Â > V_b^sâˆ’1(kâˆ’1)Â = 496305 635124 147541 989044 [i]
3. linear OA(2³, 4, F₂, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,2) or 4-cap in PG(2,2)), using
  - dual of repetition code with length 4 [i]
  - caps in base bÂ =Â 2 [i]

Mode: 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		This result only	Method
1	Linear OOA(2⁸⁴, 4106, F₂, 2, 13) (dual of [(4106, 2), 8128, 14]-NRT-code)	[i]		OOA Folding
2	Linear OOA(2⁸⁴, 2053, F₂, 4, 13) (dual of [(2053, 4), 8128, 14]-NRT-code)	[i]		OOA Folding