MinT - Information on Result #904542

Information on Result #904542

Linear OOA(81⁴⁵, 3381, F₈₁, 2, 18) (dual of [(3381, 2), 6717, 19]-NRT-code), using (u,Â u+v)-construction based on

linear OOA(81¹⁰, 100, F₈₁, 2, 9) (dual of [(100, 2), 190, 10]-NRT-code), using
- extended algebraic-geometric NRT-code AG_e(2;F,190P) [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
  - a function field by SÃ©mirat [i]
linear OOA(81³⁵, 3281, F₈₁, 2, 18) (dual of [(3281, 2), 6527, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(81³⁵, 6562, F₈₁, 18) (dual of [6562, 6527, 19]-code), using
  - discarding factors / shortening the dual code based on linear OA(81³⁵, 6563, F₈₁, 18) (dual of [6563, 6528, 19]-code), using
    - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
      1. linear OA(81³⁵, 6561, F₈₁, 18) (dual of [6561, 6526, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      2. linear OA(81³³, 6561, F₈₁, 17) (dual of [6561, 6528, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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(81⁴⁵, 1690, F₈₁, 4, 18) (dual of [(1690, 4), 6715, 19]-NRT-code)	[i]		OOA Folding