MinT - Information on Result #1119406

Information on Result #1119406

Linear OOA(3¹⁷⁸, 5033244, F₃, 5, 11) (dual of [(5033244, 5), 25166042, 12]-NRT-code), using generalized (u,Â u+v)-construction based on

linear OOA(3²⁰, 1677748, F₃, 5, 3) (dual of [(1677748, 5), 8388720, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁰, 3713024, F₃, 5, 3) (dual of [(3713024, 5), 18565100, 4]-NRT-code), using
  - appending 2 arbitrary columns [i] based on linear OOA(3²⁰, 3713024, F₃, 3, 3) (dual of [(3713024, 3), 11139052, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(3²⁰, 3713024, F₃, 2, 3) (dual of [(3713024, 2), 7426028, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3²⁰, 3713024, F₃, 3) (dual of [3713024, 3713004, 4]-code or 3713024-cap in PG(19,3)), using
        large caps in PG(u,Â 3) [i]
linear OOA(3⁴³, 1677748, F₃, 5, 5) (dual of [(1677748, 5), 8388697, 6]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3⁴³, 2391491, F₃, 5, 5) (dual of [(2391491, 5), 11957412, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(3⁴³, 4782983, F₃, 5) (dual of [4782983, 4782940, 6]-code), using
    - constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
      1. linear OA(3⁴³, 4782969, F₃, 5) (dual of [4782969, 4782926, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
      2. linear OA(3²⁹, 4782969, F₃, 4) (dual of [4782969, 4782940, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
      3. linear OA(3⁰, 14, F₃, 0) (dual of [14, 14, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 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(3¹¹⁵, 1677748, F₃, 5, 11) (dual of [(1677748, 5), 8388625, 12]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3⁹, 28, F₃, 5, 5) (dual of [(28, 5), 131, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁹, 28, F₃, 4, 5) (dual of [(28, 4), 103, 6]-NRT-code), using
      - extracting embedded OOA [i] based on digital (4, 9, 28)-net over F₃, using
        linear code embedding found in a computer search [i]
  2. linear OOA(3¹⁰⁶, 1677720, F₃, 5, 11) (dual of [(1677720, 5), 8388494, 12]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(3¹⁰⁶, 8388600, F₃, 11) (dual of [8388600, 8388494, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, large, F₃, 11) (dual of [large, large−106, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

Mode: Constructive and linear.

Optimality

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

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Linear OOA(3¹⁷⁸, 5033243, F₃, 15, 11) (dual of [(5033243, 15), 75498467, 12]-NRT-code)	[i]		OOA Stacking with Additional Row