MinT - Information on Result #734522

Information on Result #734522

Linear OA(49⁴⁹, 2506, F₄₉, 18) (dual of [2506, 2457, 19]-code), using (u,Â u+v)-construction based on

linear OA(49¹⁴, 103, F₄₉, 9) (dual of [103, 89, 10]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(49⁴, 50, F₄₉, 4) (dual of [50, 46, 5]-code or 50-arc in PG(3,49)), using
    - extended Reedâ€“Solomon code RS_e(46,49) [i]
    - algebraic-geometric code AG(F, Q+21P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using the rational function field F₄₉(x) [i]
    - algebraic-geometric code AG(F,15P) with degPÂ =Â 3 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50 (see above)
  2. linear OA(49¹⁰, 53, F₄₉, 9) (dual of [53, 43, 10]-code), using
    - constructionÂ X applied to AG(F,20P) âŠ‚ AG(F,21P) [i] based on
      1. linear OA(49⁹, 50, F₄₉, 9) (dual of [50, 41, 10]-code or 50-arc in PG(8,49)), using algebraic-geometric code AG(F,20P) with degPÂ =Â 2 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50 (see above)
      2. linear OA(49⁷, 50, F₄₉, 7) (dual of [50, 43, 8]-code or 50-arc in PG(6,49)), using algebraic-geometric code AG(F,21P) with degPÂ =Â 2 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50 (see above)
      3. linear OA(49¹, 3, F₄₉, 1) (dual of [3, 2, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
linear OA(49³⁵, 2403, F₄₉, 18) (dual of [2403, 2368, 19]-code), using
- constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
  1. linear OA(49³⁵, 2401, F₄₉, 18) (dual of [2401, 2366, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
  2. linear OA(49³³, 2401, F₄₉, 17) (dual of [2401, 2368, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
  3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 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(49⁴⁹, 1253, F₄₉, 2, 18) (dual of [(1253, 2), 2457, 19]-NRT-code)	[i]		OOA Folding
2	Linear OOA(49⁴⁹, 835, F₄₉, 3, 18) (dual of [(835, 3), 2456, 19]-NRT-code)	[i]		OOA Folding