MinT - Information on Result #2266287

Information on Result #2266287

Linear OA(3¹³⁷, 782, F₃, 31) (dual of [782, 645, 32]-code), using 1 times code embedding in larger space based on linear OA(3¹³⁶, 781, F₃, 31) (dual of [781, 645, 32]-code), using

constructionÂ XX applied to C₁Â =Â C([722,19]), C₂Â =Â C([0,24]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,19]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([722,24]) [i] based on
1. linear OA(3¹⁰³, 728, F₃, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,19}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
2. linear OA(3⁹⁷, 728, F₃, 25) (dual of [728, 631, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [0,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
3. linear OA(3¹²¹, 728, F₃, 31) (dual of [728, 607, 32]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,24}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
4. linear OA(3⁷⁹, 728, F₃, 20) (dual of [728, 649, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
5. linear OA(3⁸, 28, F₃, 5) (dual of [28, 20, 6]-code), using
  - dual code (with bound on d by constructionÂ Y1) [i] based on
    1. linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code), using
      - contraction [i] based on linear OA(3⁴⁸, 56, F₃, 29) (dual of [56, 8, 30]-code), using the BCH-code C(I) with length 56Â | 3⁶âˆ’1, defining interval IÂ =Â {0,1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. nonexistence of linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), because
      - 2 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
6. linear OA(3⁷, 25, F₃, 4) (dual of [25, 18, 5]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁷, 26, F₃, 4) (dual of [26, 19, 5]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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(3¹³⁷, 391, F₃, 2, 31) (dual of [(391, 2), 645, 32]-NRT-code)	[i]		OOA Folding