MinT - Information on Result #661467

Information on Result #661467

Linear OA(2²³³, 345, F₂, 67) (dual of [345, 112, 68]-code), using concatenation of two codes based on

linear OA(16⁴¹, 69, F₁₆, 33) (dual of [69, 28, 34]-code), using
- constructionÂ X applied to AG(F,30P) âŠ‚ AG(F,33P) [i] based on
  1. linear OA(16³⁹, 64, F₁₆, 33) (dual of [64, 25, 34]-code), using algebraic-geometric code AG(F,30P) [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(16³⁶, 64, F₁₆, 30) (dual of [64, 28, 31]-code), using algebraic-geometric code AG(F,33P) [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65 (see above)
  3. linear OA(16², 5, F₁₆, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,16)), using
    - discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
      - Reedâ€“Solomon code RS(14,16) [i]
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]

Mode: Constructive and linear.

None.

The following results depend on this result:

	Result		Method
1	Linear OA(2²³³, 345, F₂, 66) (dual of [345, 112, 67]-code)	[i]	Strength Reduction
2	Linear OA(2²³⁴, 346, F₂, 67) (dual of [346, 112, 68]-code)	[i]	Code Embedding in Larger Space
3	Linear OA(2²³⁵, 347, F₂, 67) (dual of [347, 112, 68]-code)	[i]
4	Linear OA(2²³⁶, 348, F₂, 67) (dual of [348, 112, 68]-code)	[i]
5	Linear OA(2²³⁷, 349, F₂, 67) (dual of [349, 112, 68]-code)	[i]
6	Linear OA(2²³⁸, 350, F₂, 67) (dual of [350, 112, 68]-code)	[i]
7	Linear OA(2²³², 344, F₂, 66) (dual of [344, 112, 67]-code)	[i]	Truncation
8	Linear OOA(2²³³, 172, F₂, 2, 67) (dual of [(172, 2), 111, 68]-NRT-code)	[i]	OOA Folding
9	Linear OOA(2²³³, 115, F₂, 3, 67) (dual of [(115, 3), 112, 68]-NRT-code)	[i]	OOA Folding