MinT - Information on Result #1084735

Information on Result #1084735

Linear OOA(81⁸⁰, large, F₈₁, 4, 12), using generalized (u,Â u+v)-construction based on

linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 4, 0) for arbitrarily large s, using
  - OOA with only one run / dual of NRT-code without redundancy [i]
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 132861, F₈₁, 4, 0) (dual of [(132861, 4), 531444, 1]-NRT-code) (see above)
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 4, 1) for arbitrarily large s, using
  - appending 3 arbitrary columns [i] based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
    - parity-check code [i]
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 132861, F₈₁, 4, 1) (dual of [(132861, 4), 531443, 2]-NRT-code) (see above)
linear OOA(81⁴, 132861, F₈₁, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81⁴, 538084, F₈₁, 4, 2) (dual of [(538084, 4), 2152332, 3]-NRT-code), using
  - appending 2 arbitrary columns [i] based on linear OOA(81⁴, 538084, F₈₁, 2, 2) (dual of [(538084, 2), 1076164, 3]-NRT-code), using
    - appending kth column [i] based on linear OA(81⁴, 538084, F₈₁, 2) (dual of [538084, 538080, 3]-code), using
      - Hamming code H(4,81) [i]
linear OOA(81⁴, 132861, F₈₁, 4, 2) (dual of [(132861, 4), 531440, 3]-NRT-code) (see above)
linear OOA(81⁶, 132861, F₈₁, 4, 3) (dual of [(132861, 4), 531438, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81⁶, 538248, F₈₁, 4, 3) (dual of [(538248, 4), 2152986, 4]-NRT-code), using
  - appending 1 arbitrary column [i] based on linear OOA(81⁶, 538248, F₈₁, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(81⁶, 538248, F₈₁, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
linear OOA(81¹⁰, 132861, F₈₁, 4, 4) (dual of [(132861, 4), 531434, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81¹⁰, 265722, F₈₁, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
    - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
      1. linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
      2. linear OA(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
      3. linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 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(81¹⁶, 132861, F₈₁, 4, 6) (dual of [(132861, 4), 531428, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81¹⁶, 177148, F₈₁, 4, 6) (dual of [(177148, 4), 708576, 7]-NRT-code), using
  - OA 3-folding and stacking [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
    - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
      1. linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      2. linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
      3. linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)
linear OOA(81³⁴, 132861, F₈₁, 4, 12) (dual of [(132861, 4), 531410, 13]-NRT-code), using
- OOA 4-folding [i] based on linear OA(81³⁴, 531444, F₈₁, 12) (dual of [531444, 531410, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
    1. linear OA(81³⁴, 531441, F₈₁, 12) (dual of [531441, 531407, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(81³¹, 531441, F₈₁, 11) (dual of [531441, 531410, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)

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(81⁸⁰, large, F₈₁, 4, 11)	[i]	Strength Reduction for OOAs
2	Linear OOA(81⁸⁰, large, F₈₁, 4, 10)	[i]
3	Linear OOA(81⁸⁰, large, F₈₁, 4, 9)	[i]
4	Linear OOA(81⁸⁰, large, F₈₁, 4, 8)	[i]
5	Linear OOA(81⁸⁰, large, F₈₁, 4, 7)	[i]
6	Linear OOA(81⁸⁰, large, F₈₁, 4, 6)	[i]
7	Linear OOA(81⁸⁰, large, F₈₁, 4, 5)	[i]
8	Linear OOA(81⁸⁰, large, F₈₁, 4, 4)	[i]
9	Linear OOA(81⁸⁰, large, F₈₁, 4, 3)	[i]
10	Linear OOA(81⁸¹, large, F₈₁, 4, 12)	[i]	OOA Duplication
11	Linear OOA(81⁸², large, F₈₁, 4, 12)	[i]	OOA Duplication