MinT - Information on Result #1567349

Information on Result #1567349

Linear OOA(256³⁴, 65795, F₂₅₆, 3, 14) (dual of [(65795, 3), 197351, 15]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OA(256³⁴, 65795, F₂₅₆, 14) (dual of [65795, 65761, 15]-code), using

(u,Â u+v)-construction [i] based on
1. linear OA(256⁷, 257, F₂₅₆, 7) (dual of [257, 250, 8]-code or 257-arc in PG(6,256)), using
  - extended Reedâ€“Solomon code RS_e(250,256) [i]
  - the expurgated narrow-sense BCH-code C(I) with length 257Â | 256²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
  - algebraic-geometric code AG(F, Q+123P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using the rational function field F₂₅₆(x) [i]
  - algebraic-geometric code AG(F,83P) with degPÂ =Â 3 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
  - algebraic-geometric code AG(F, Q+49P) with degQ = 4 and degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
2. linear OA(256²⁷, 65538, F₂₅₆, 14) (dual of [65538, 65511, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. linear OA(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(256²⁵, 65536, F₂₅₆, 13) (dual of [65536, 65511, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

Mode: 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(256³⁴, 32897, F₂₅₆, 5, 14) (dual of [(32897, 5), 164451, 15]-NRT-code)	[i]		OOA Folding
2	Linear OOA(256³⁴, 32897, F₂₅₆, 6, 14) (dual of [(32897, 6), 197348, 15]-NRT-code)	[i]		OOA Folding