MinT - Information on Result #683125

Information on Result #683125

Linear OA(5¹²², 3176, F₅, 27) (dual of [3176, 3054, 28]-code), using constructionÂ X applied to C_e(26) âŠ‚ C_e(17) based on

linear OA(5¹⁰⁶, 3125, F₅, 27) (dual of [3125, 3019, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
linear OA(5⁷¹, 3125, F₅, 18) (dual of [3125, 3054, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
linear OA(5¹⁶, 51, F₅, 8) (dual of [51, 35, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁶, 52, F₅, 8) (dual of [52, 36, 9]-code), using
  - trace code [i] based on linear OA(25⁸, 26, F₂₅, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,25)), using
    - extended Reedâ€“Solomon code RS_e(18,25) [i]
    - algebraic-geometric code AG(F, Q+7P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using the rational function field F₂₅(x) [i]
    - algebraic-geometric code AG(F, Q+5P) with degQ = 2 and degPÂ =Â 3 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26 (see above)

Mode: Constructive and linear.

None.

The following results depend on this result:

	Result		Method
1	Linear OA(5¹²³, 3177, F₅, 27) (dual of [3177, 3054, 28]-code)	[i]	Code Embedding in Larger Space
2	Linear OA(5¹²⁴, 3178, F₅, 27) (dual of [3178, 3054, 28]-code)	[i]	Code Embedding in Larger Space
3	Linear OOA(5¹²², 1588, F₅, 2, 27) (dual of [(1588, 2), 3054, 28]-NRT-code)	[i]	OOA Folding
4	Linear OOA(5¹²², 1058, F₅, 3, 27) (dual of [(1058, 3), 3052, 28]-NRT-code)	[i]	OOA Folding