MinT - Information on Result #904687

Information on Result #904687

Linear OOA(81⁶⁸, 3431, F₈₁, 2, 26) (dual of [(3431, 2), 6794, 27]-NRT-code), using (u,Â u+v)-construction based on

linear OOA(81¹⁷, 150, F₈₁, 2, 13) (dual of [(150, 2), 283, 14]-NRT-code), using
- extended algebraic-geometric NRT-code AG_e(2;F,286P) [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
  - a function field by SÃ©mirat [i]
linear OOA(81⁵¹, 3281, F₈₁, 2, 26) (dual of [(3281, 2), 6511, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(81⁵¹, 6562, F₈₁, 26) (dual of [6562, 6511, 27]-code), using
  - discarding factors / shortening the dual code based on linear OA(81⁵¹, 6563, F₈₁, 26) (dual of [6563, 6512, 27]-code), using
    - constructionÂ X applied to C_e(25) âŠ‚ C_e(24) [i] based on
      1. linear OA(81⁵¹, 6561, F₈₁, 26) (dual of [6561, 6510, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      2. linear OA(81⁴⁹, 6561, F₈₁, 25) (dual of [6561, 6512, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      3. linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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		This result only	Method
1	Linear OOA(81⁶⁸, 1715, F₈₁, 4, 26) (dual of [(1715, 4), 6792, 27]-NRT-code)	[i]		OOA Folding