MinT - Information on Result #1222056

Information on Result #1222056

Digital (76, 87, 3375904)-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 (55, 66, 1687952)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 15, 10232)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27¹⁵, 10232, F₂₇, 6, 5) (dual of [(10232, 6), 61377, 6]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(27¹⁵, 10233, F₂₇, 2, 5) (dual of [(10233, 2), 20451, 6]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 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, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27⁰, 379, F₂₇, 2, 0) (dual of [(379, 2), 758, 1]-NRT-code) (see above)
        linear OOA(27¹, 379, F₂₇, 2, 1) (dual of [(379, 2), 757, 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, 379, F₂₇, 2, 1) (dual of [(379, 2), 757, 2]-NRT-code) (see above)
        linear OOA(27¹, 379, F₂₇, 2, 1) (dual of [(379, 2), 757, 2]-NRT-code) (see above)
        linear OOA(27³, 379, F₂₇, 2, 2) (dual of [(379, 2), 755, 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(27⁹, 379, F₂₇, 2, 5) (dual of [(379, 2), 749, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        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(27⁷, 351, F₂₇, 2, 5) (dual of [(351, 2), 695, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(27⁷, 702, F₂₇, 5) (dual of [702, 695, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (40, 51, 1677720)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [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 (97, 108, 6751808)-net over F₂₇	[i]		(u,Â u+v)-Construction for Nets