MinT - Information on Result #907033

Information on Result #907033

Linear OOA(81³³, large, F₈₁, 2, 7), using generalized (u,Â u+v)-construction based on

linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 2, 0) for arbitrarily large s, using
  - OOA with only one run / dual of NRT-code without redundancy [i]
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81⁰, 265722, F₈₁, 2, 0) (dual of [(265722, 2), 531444, 1]-NRT-code) (see above)
linear OOA(81¹, 265722, F₈₁, 2, 1) (dual of [(265722, 2), 531443, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 2, 1) for arbitrarily large s, using
  - appending 1 arbitrary column [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, 265722, F₈₁, 2, 1) (dual of [(265722, 2), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 265722, F₈₁, 2, 1) (dual of [(265722, 2), 531443, 2]-NRT-code) (see above)
linear OOA(81¹, 265722, F₈₁, 2, 1) (dual of [(265722, 2), 531443, 2]-NRT-code) (see above)
linear OOA(81⁴, 265722, F₈₁, 2, 2) (dual of [(265722, 2), 531440, 3]-NRT-code), using
- discarding factors / shortening the dual code 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(81⁶, 265722, F₈₁, 2, 3) (dual of [(265722, 2), 531438, 4]-NRT-code), using
- discarding factors / shortening the dual code 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(81¹⁹, 265722, F₈₁, 2, 7) (dual of [(265722, 2), 531425, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(81¹⁹, 531444, F₈₁, 7) (dual of [531444, 531425, 8]-code), using
  - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
    1. linear OA(81¹⁹, 531441, F₈₁, 7) (dual of [531441, 531422, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. 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]
    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]

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(81³³, large, F₈₁, 2, 6)	[i]	Strength Reduction for OOAs
2	Linear OOA(81³³, large, F₈₁, 2, 5)	[i]
3	Linear OOA(81³³, large, F₈₁, 2, 4)	[i]
4	Linear OOA(81³³, large, F₈₁, 2, 3)	[i]
5	Linear OOA(81³⁵, large, F₈₁, 2, 7)	[i]	NRT-Code Embedding in Larger Space
6	Linear OOA(81³⁷, large, F₈₁, 2, 7)	[i]
7	Linear OOA(81³⁹, large, F₈₁, 2, 7)	[i]
8	Linear OOA(81⁴¹, large, F₈₁, 2, 7)	[i]
9	Linear OOA(81⁴³, large, F₈₁, 2, 7)	[i]
10	Linear OOA(81⁴⁵, large, F₈₁, 2, 7)	[i]
11	Linear OOA(81⁴⁷, large, F₈₁, 2, 7)	[i]
12	Linear OOA(81⁴⁹, large, F₈₁, 2, 7)	[i]
13	Linear OOA(81⁵¹, large, F₈₁, 2, 7)	[i]
14	Linear OOA(81⁵³, large, F₈₁, 2, 7)	[i]
15	Linear OOA(81⁵⁵, large, F₈₁, 2, 7)	[i]
16	Linear OOA(81⁵⁷, large, F₈₁, 2, 7)	[i]
17	Linear OOA(81⁵⁹, large, F₈₁, 2, 7)	[i]
18	Linear OOA(81⁶¹, large, F₈₁, 2, 7)	[i]
19	Linear OOA(81⁶³, large, F₈₁, 2, 7)	[i]
20	Linear OOA(81⁶⁵, large, F₈₁, 2, 7)	[i]
21	Linear OOA(81⁶⁷, large, F₈₁, 2, 7)	[i]
22	Linear OOA(81⁶⁹, large, F₈₁, 2, 7)	[i]
23	Linear OOA(81⁷¹, large, F₈₁, 2, 7)	[i]
24	Linear OOA(81⁷³, large, F₈₁, 2, 7)	[i]
25	Linear OOA(81⁷⁵, large, F₈₁, 2, 7)	[i]
26	Linear OOA(81⁷⁷, large, F₈₁, 2, 7)	[i]
27	Linear OOA(81⁷⁹, large, F₈₁, 2, 7)	[i]
28	Linear OOA(81⁸¹, large, F₈₁, 2, 7)	[i]
29	Linear OOA(81³⁴, large, F₈₁, 2, 7)	[i]	OOA Duplication