MinT - Information on Result #1221991

Information on Result #1221991

Digital (73, 83, 3396262)-net over F₂₇, using (u,Â u+v)-construction based on

digital (16, 21, 4194301)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27²¹, 4194301, F₂₇, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(27²¹, large, F₂₇, 5) (dual of [large, large−21, 6]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
digital (52, 62, 1698131)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 16, 20411)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27¹⁶, 20411, F₂₇, 6, 5) (dual of [(20411, 6), 122450, 6]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(27¹⁶, 20412, F₂₇, 2, 5) (dual of [(20412, 2), 40808, 6]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(27⁰, s, F₂₇, 2, 0) for arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27⁰, 756, F₂₇, 2, 0) (dual of [(756, 2), 1512, 1]-NRT-code) (see above)
        linear OOA(27¹, 756, F₂₇, 2, 1) (dual of [(756, 2), 1511, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(27¹, s, F₂₇, 2, 1) for arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
        linear OOA(27¹, 756, F₂₇, 2, 1) (dual of [(756, 2), 1511, 2]-NRT-code) (see above)
        linear OOA(27¹, 756, F₂₇, 2, 1) (dual of [(756, 2), 1511, 2]-NRT-code) (see above)
        linear OOA(27³, 756, F₂₇, 2, 2) (dual of [(756, 2), 1509, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(27³, 757, F₂₇, 2, 2) (dual of [(757, 2), 1511, 3]-NRT-code), using
        appending kth column [i] based on linear OA(27³, 757, F₂₇, 2) (dual of [757, 754, 3]-code), using
        Hamming code H(3,27) [i]
        linear OOA(27¹⁰, 756, F₂₇, 2, 5) (dual of [(756, 2), 1502, 6]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(27⁰, s, F₂₇, 2, 0) for arbitrarily large s (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27⁰, 28, F₂₇, 2, 0) (dual of [(28, 2), 56, 1]-NRT-code) (see above)
        linear OOA(27¹, 28, F₂₇, 2, 1) (dual of [(28, 2), 55, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(27¹, s, F₂₇, 2, 1) for arbitrarily large s (see above)
        linear OOA(27¹, 28, F₂₇, 2, 1) (dual of [(28, 2), 55, 2]-NRT-code) (see above)
        linear OOA(27¹, 28, F₂₇, 2, 1) (dual of [(28, 2), 55, 2]-NRT-code) (see above)
        linear OOA(27², 28, F₂₇, 2, 2) (dual of [(28, 2), 54, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;54,27) [i]
        linear OOA(27⁵, 28, F₂₇, 2, 5) (dual of [(28, 2), 51, 6]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;51,27) [i]
  2. digital (36, 46, 1677720)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁴⁶, 1677720, F₂₇, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(27⁴⁶, 8388600, F₂₇, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁴⁶, large, F₂₇, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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, t and s.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Digital (94, 104, 6792524)-net over F₂₇	[i]		(u,Â u+v)-Construction for Nets