MinT - Information on Result #1022144

Information on Result #1022144

Linear OOA(64⁵¹, large, F₆₄, 3, 8), using generalized (u,Â u+v)-construction based on

linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁰, s, F₆₄, 3, 0) for arbitrarily large s, using
  - OOA with only one run / dual of NRT-code without redundancy [i]
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64⁰, 131073, F₆₄, 3, 0) (dual of [(131073, 3), 393219, 1]-NRT-code) (see above)
linear OOA(64¹, 131073, F₆₄, 3, 1) (dual of [(131073, 3), 393218, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64¹, s, F₆₄, 3, 1) for arbitrarily large s, using
  - appending 2 arbitrary columns [i] based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
    - parity-check code [i]
linear OOA(64¹, 131073, F₆₄, 3, 1) (dual of [(131073, 3), 393218, 2]-NRT-code) (see above)
linear OOA(64¹, 131073, F₆₄, 3, 1) (dual of [(131073, 3), 393218, 2]-NRT-code) (see above)
linear OOA(64¹, 131073, F₆₄, 3, 1) (dual of [(131073, 3), 393218, 2]-NRT-code) (see above)
linear OOA(64⁴, 131073, F₆₄, 3, 2) (dual of [(131073, 3), 393215, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁴, 266305, F₆₄, 3, 2) (dual of [(266305, 3), 798911, 3]-NRT-code), using
  - appending 1 arbitrary column [i] based on linear OOA(64⁴, 266305, F₆₄, 2, 2) (dual of [(266305, 2), 532606, 3]-NRT-code), using
    - appending kth column [i] based on linear OA(64⁴, 266305, F₆₄, 2) (dual of [266305, 266301, 3]-code), using
      - Hamming code H(4,64) [i]
linear OOA(64⁴, 131073, F₆₄, 3, 2) (dual of [(131073, 3), 393215, 3]-NRT-code) (see above)
linear OOA(64¹⁰, 131073, F₆₄, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(64¹⁰, 262146, F₆₄, 4) (dual of [262146, 262136, 5]-code), using
  - discarding factors / shortening the dual code based on linear OA(64¹⁰, 262147, F₆₄, 4) (dual of [262147, 262137, 5]-code), using
    - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
      1. linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
      2. linear OA(64⁷, 262144, F₆₄, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
      3. linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
linear OOA(64²⁹, 2796201, F₆₄, 3, 8) (dual of [(2796201, 3), 8388574, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Linear OOA(64⁵¹, large, F₆₄, 3, 7)	[i]	Strength Reduction for OOAs
2	Linear OOA(64⁵¹, large, F₆₄, 3, 6)	[i]
3	Linear OOA(64⁵¹, large, F₆₄, 3, 5)	[i]
4	Linear OOA(64⁵¹, large, F₆₄, 3, 4)	[i]
5	Linear OOA(64⁵¹, large, F₆₄, 3, 3)	[i]