MinT - Information on Result #904581

Information on Result #904581

Linear OOA(81⁶⁷, 265840, F₈₁, 2, 19) (dual of [(265840, 2), 531613, 20]-NRT-code), using (u,Â u+v)-construction based on

linear OOA(81¹¹, 116, F₈₁, 2, 9) (dual of [(116, 2), 221, 10]-NRT-code), using
- extended algebraic-geometric NRT-code AG_e(2;F,222P) [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
  - a function field by SÃ©mirat [i]
linear OOA(81⁵⁶, 265724, F₈₁, 2, 19) (dual of [(265724, 2), 531392, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(81⁵⁶, 531448, F₈₁, 19) (dual of [531448, 531392, 20]-code), using
  - discarding factors / shortening the dual code based on linear OA(81⁵⁶, 531449, F₈₁, 19) (dual of [531449, 531393, 20]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
      1. linear OA(81⁵⁵, 531442, F₈₁, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(81⁴⁹, 531442, F₈₁, 17) (dual of [531442, 531393, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      3. linear OA(81¹, 7, F₈₁, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 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		This result only	Method
1	Linear OOA(81⁶⁷, 132920, F₈₁, 4, 19) (dual of [(132920, 4), 531613, 20]-NRT-code)	[i]		OOA Folding