MinT - Information on Result #873334

Information on Result #873334

Linear OOA(49⁵¹, 2882427, F₄₉, 2, 12) (dual of [(2882427, 2), 5764803, 13]-NRT-code), using OOA 2-folding based on linear OA(49⁵¹, 5764854, F₄₉, 12) (dual of [5764854, 5764803, 13]-code), using

discarding factors / shortening the dual code based on linear OA(49⁵¹, 5764855, F₄₉, 12) (dual of [5764855, 5764804, 13]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(49⁶, 50, F₄₉, 6) (dual of [50, 44, 7]-code or 50-arc in PG(5,49)), using
    - extended Reedâ€“Solomon code RS_e(44,49) [i]
    - algebraic-geometric code AG(F, Q+20P) 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, Q+13P) with degQ = 4 and degPÂ =Â 3 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50 (see above)
    - algebraic-geometric code AG(F, Q+8P) with degQ = 3 and degPÂ =Â 5 [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50 (see above)
  2. linear OA(49⁴⁵, 5764805, F₄₉, 12) (dual of [5764805, 5764760, 13]-code), using
    - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
      1. linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      2. linear OA(49⁴¹, 5764801, F₄₉, 11) (dual of [5764801, 5764760, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 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(7¹⁰², 5764854, F₇, 2, 12) (dual of [(5764854, 2), 11529606, 13]-NRT-code)	[i]		Trace Code for OOAs