MinT - Information on Result #988715

Information on Result #988715

Linear OOA(49⁵³, 1921621, F₄₉, 3, 12) (dual of [(1921621, 3), 5764810, 13]-NRT-code), using OOA 3-folding based on linear OA(49⁵³, 5764863, F₄₉, 12) (dual of [5764863, 5764810, 13]-code), using

(u,Â u+v)-construction [i] based on
1. linear OA(49⁷, 53, F₄₉, 6) (dual of [53, 46, 7]-code), using
  - constructionÂ X applied to AG(F, Q+20P) âŠ‚ AG(F, Q+21P) [i] based on
    1. linear OA(49⁶, 50, F₄₉, 6) (dual of [50, 44, 7]-code or 50-arc in PG(5,49)), using 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]
    2. linear OA(49⁴, 50, F₄₉, 4) (dual of [50, 46, 5]-code or 50-arc in PG(3,49)), using 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 (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]
2. linear OA(49⁴⁶, 5764810, F₄₉, 12) (dual of [5764810, 5764764, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(9) [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(49³⁷, 5764801, F₄₉, 10) (dual of [5764801, 5764764, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(49¹, 9, F₄₉, 1) (dual of [9, 8, 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 (see above)

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(7¹⁰⁶, 3843242, F₇, 3, 12) (dual of [(3843242, 3), 11529620, 13]-NRT-code)	[i]		Trace Code for OOAs