MinT - Information on Result #2295650

Information on Result #2295650

Linear OA(2¹⁹², 329, F₂, 51) (dual of [329, 137, 52]-code), using adding a parity check bit based on linear OA(2¹⁹¹, 328, F₂, 50) (dual of [328, 137, 51]-code), using

constructionÂ X with VarÅ¡amov bound [i] based on
1. linear OA(2¹⁸⁸, 324, F₂, 50) (dual of [324, 136, 51]-code), using
  - 1 times truncation [i] based on linear OA(2¹⁸⁹, 325, F₂, 51) (dual of [325, 136, 52]-code), using
    - concatenation of two codes [i] based on
      1. linear OA(16³¹, 65, F₁₆, 25) (dual of [65, 34, 26]-code), using
        extended algebraic-geometric code AG_e(F,39P) [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
      2. linear OA(2¹, 5, F₂, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
2. linear OA(2¹⁸⁸, 325, F₂, 47) (dual of [325, 137, 48]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 2¹⁸⁸Â > V_b^sâˆ’1(kâˆ’1)Â = 231 778270 877159 784183 087333 887721 106612 103677 087018 507936 [i]
3. linear OA(2², 3, F₂, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,2)), using
  - dual of repetition code with length 3 [i]
  - Hamming code H(2,2) [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 OA(2¹⁹², 329, F₂, 50) (dual of [329, 137, 51]-code)	[i]		Strength Reduction
2	Linear OA(2¹⁹³, 330, F₂, 51) (dual of [330, 137, 52]-code)	[i]		Code Embedding in Larger Space