MinT - Information on Result #1084750

Information on Result #1084750

Linear OOA(81³³, large, F₈₁, 4, 7), using generalized (u,Â u+v)-construction based on

linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 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, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81⁰, 177147, F₈₁, 4, 0) (dual of [(177147, 4), 708588, 1]-NRT-code) (see above)
linear OOA(81¹, 177147, F₈₁, 4, 1) (dual of [(177147, 4), 708587, 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, 177147, F₈₁, 4, 1) (dual of [(177147, 4), 708587, 2]-NRT-code) (see above)
linear OOA(81¹, 177147, F₈₁, 4, 1) (dual of [(177147, 4), 708587, 2]-NRT-code) (see above)
linear OOA(81¹, 177147, F₈₁, 4, 1) (dual of [(177147, 4), 708587, 2]-NRT-code) (see above)
linear OOA(81⁴, 177147, F₈₁, 4, 2) (dual of [(177147, 4), 708584, 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(81⁶, 177147, F₈₁, 4, 3) (dual of [(177147, 4), 708582, 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(81¹⁹, 177147, F₈₁, 4, 7) (dual of [(177147, 4), 708569, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(81¹⁹, 531442, F₈₁, 7) (dual of [531442, 531423, 8]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [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, 4, 6)	[i]	Strength Reduction for OOAs
2	Linear OOA(81³³, large, F₈₁, 4, 5)	[i]
3	Linear OOA(81³³, large, F₈₁, 4, 4)	[i]
4	Linear OOA(81³³, large, F₈₁, 4, 3)	[i]
5	Linear OOA(81³⁷, large, F₈₁, 4, 7)	[i]	NRT-Code Embedding in Larger Space
6	Linear OOA(81⁴¹, large, F₈₁, 4, 7)	[i]
7	Linear OOA(81⁴⁵, large, F₈₁, 4, 7)	[i]
8	Linear OOA(81⁴⁹, large, F₈₁, 4, 7)	[i]
9	Linear OOA(81⁵³, large, F₈₁, 4, 7)	[i]
10	Linear OOA(81⁶⁵, large, F₈₁, 4, 7)	[i]
11	Linear OOA(81⁶⁹, large, F₈₁, 4, 7)	[i]
12	Linear OOA(81⁸¹, large, F₈₁, 4, 7)	[i]
13	Linear OOA(81³⁴, large, F₈₁, 4, 7)	[i]	OOA Duplication
14	Linear OOA(81³⁵, large, F₈₁, 4, 7)	[i]	OOA Duplication