MinT - Information on Result #735482

Information on Result #735482

Linear OA(81²⁹, 531526, F₈₁, 9) (dual of [531526, 531497, 10]-code), using (u,Â u+v)-construction based on

linear OA(81⁴, 82, F₈₁, 4) (dual of [82, 78, 5]-code or 82-arc in PG(3,81)), using
- extended Reedâ€“Solomon code RS_e(78,81) [i]
linear OA(81²⁵, 531444, F₈₁, 9) (dual of [531444, 531419, 10]-code), using
- constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
  1. linear OA(81²⁵, 531441, F₈₁, 9) (dual of [531441, 531416, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  2. linear OA(81²², 531441, F₈₁, 8) (dual of [531441, 531419, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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.

None.

The following results depend on this result:

	Result		Method
1	Linear OOA(81²⁹, 531526, F₈₁, 2, 9) (dual of [(531526, 2), 1063023, 10]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(81²⁹, 531526, F₈₁, 3, 9) (dual of [(531526, 3), 1594549, 10]-NRT-code)	[i]
3	Linear OOA(81²⁹, 531526, F₈₁, 4, 9) (dual of [(531526, 4), 2126075, 10]-NRT-code)	[i]
4	Digital (20, 29, 531526)-net over F₈₁	[i]
5	Linear OOA(81²⁹, 265763, F₈₁, 2, 9) (dual of [(265763, 2), 531497, 10]-NRT-code)	[i]	OOA Folding
6	Linear OOA(81²⁹, 177175, F₈₁, 3, 9) (dual of [(177175, 3), 531496, 10]-NRT-code)	[i]	OOA Folding