MinT - Information on Result #642460

Information on Result #642460

Linear OA(25⁷⁸, 81, F₂₅, 73) (dual of [81, 3, 74]-code), using juxtaposition based on

linear OA(25²³, 26, F₂₅, 23) (dual of [26, 3, 24]-code or 26-arc in PG(22,25)), using
- extended Reedâ€“Solomon code RS_e(3,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26Â | 25²âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
- algebraic-geometric code AG(F,1P) with degPÂ =Â 2 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using the rational function field F₂₅(x) [i]
- algebraic-geometric code AG(F, Q+0P) with degQ = 2 and degPÂ =Â 3 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26 (see above)
linear OA(25⁵², 55, F₂₅, 49) (dual of [55, 3, 50]-code), using
- juxtaposition [i] based on
  1. linear OA(25²³, 26, F₂₅, 23) (dual of [26, 3, 24]-code or 26-arc in PG(22,25)) (see above)
  2. linear OA(25²⁶, 29, F₂₅, 25) (dual of [29, 3, 26]-code), using
    - constructionÂ X applied to C₁ âŠ‚ C₀ [i] based on
      1. linear OA(25²⁵, 26, F₂₅, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,25)), using code C₁ for uÂ =Â 2 by de Boer and Brouwer [i]
      2. linear OA(25²³, 26, F₂₅, 23) (dual of [26, 3, 24]-code or 26-arc in PG(22,25)), using code C₀ for uÂ =Â 2 by de Boer and Brouwer [i]
      3. linear OA(25¹, 3, F₂₅, 1) (dual of [3, 2, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 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 OA(25⁷⁸, 81, F₂₅, 72) (dual of [81, 3, 73]-code)	[i]		Strength Reduction
2	Linear OA(25¹⁰⁴, 107, F₂₅, 97) (dual of [107, 3, 98]-code)	[i]		Juxtaposition