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New optimal error-correcting codes for crosstalk avoidance in on-chip data buses

  • * Corresponding author: Xiande Zhang

    * Corresponding author: Xiande Zhang

The first author is supported by the Chinese Scholarship Council at USTC, China. The second author is supported by NSFC grant 11771419, and by the Fundamental Research Funds for the Central Universities

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  • Codes that simultaneously provide for low power dissipation, cross-talk avoidance, and error correction in the ultra deep submicron/nanometer VLSI fabrication, were recently introduced by Chee et al. in 2015. Such codes were revealed to be closely related to balanced sampling plans avoiding adjacent units, which are widely used in the statistical design of experiments. In this paper, we construct a new family of optimal codes with such properties, by determining the maximum size of packing sampling plans avoiding certain units.

    Mathematics Subject Classification: Primary: 05B07, 05B40; Secondary: 94B25.

    Citation:

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  • Figure 1.  Leave graphs of LPSA$ (6n,3;2) $ having $ 3n $ edges

    Figure 2.  Points of the hole are denoted by $ \bullet $ and others by $ \circ $

    Figure 3.  Points of the hole are denoted by $ \bullet $ and others by $ \circ $

    Table 1.  Types of worst crosstalk couplings

    Type-Ⅰ Type-Ⅱ Type-Ⅲ Type-Ⅳ
    $ 0\longleftrightarrow1 $ $ 001\longleftrightarrow010 $ $ 010\longleftrightarrow101 $
    $ 001\longleftrightarrow110 $ $ 010\longleftrightarrow100 $
    $ 011\longleftrightarrow100 $ $ 011\longleftrightarrow101 $
    $ 101\longleftrightarrow110 $
    Single wire undergoes transition. Adjacent wires maintain previous states Center wire in opposite transition to an adjacent wire. The other wire in same transition as center wire Center wire in opposite transition to an adjacent wire. The other wire maintains previous state All three adjacent wires undergo opposite transitions
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    Table 2.  Upper bounds and leave graphs of CPSAs and LPSAs excluding edges within distance two

    Sampling plans Upper bounds Leave graphs
    CPSA$ (6n,3;2) $ $ 6n(n-1) $ A perfect matching
    CPSA$ (6n+1,3;2) $ $ n(6n-3)-2 $ A cycle of length four
    LPSA$ (6n,3;2) $ $ 6n(n-1)+1 $ see Fig. 1
    LPSA$ (6n+1,3;2) $ $ n(6n-3) $ A single edge
     | Show Table
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    Table 3.  Existence results of small orders

    $ n $ $ 24 $ $ 25 $ $ 30 $ $ 31 $ $ 36 $ $ 37 $ $ 48 $ $ 49 $
    $ B^{\circ}(n,3;2) $ $ - $ $ 82 $ $ - $ $ 133 $ $ - $ $ 196 $ $ 336 $ $ 358 $
    $ B(n,3;2) $ $ 73 $ $ 84 $ $ 121 $ $ 135 $ $ 181 $ $ 198 $ $ 337 $ $ 360 $
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    Table 4.  Parameters for proof of Lemma 4.6

    n=3g+t-1 g t s
    72s=3(24(s-1)+18)+18 24(s-1)+18 19 $s\geq 1, \ s\neq 1$
    72s+24=3(24(s-1)+18)+42 24(s-1)+18 43 $s\geq 1, \ s\neq 1,2,3$
    72s+48=3(24s+12)+12 24s+12 13 $s\geq 1$
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    Table 5.  Parameters for proof of Lemma 4.8

    $n\quad \ \ \ \ \ \ =3g+t$ $g$ $t$ $s$
    $72s+1 \ =3(24(s-1)+18)+19$ $24(s-1)+18$ $19$ $s\geq 1,\ s\neq 1$
    $72s+7 \ =3(24s)+7$ $24s$ $7$ $s\geq 1$
    $72s+13=3(24(s-1)+18)+31$ $24(s-1)+18$ $31$ $s\geq 1,\ s\neq 1,2,3$
    $72s+19=3(24s+6)+1$ $24s+6$ $1$ $s\geq 1$
    $72s+25=3(24(s-1)+18)+43$ $24(s-1)+18$ $43$ $s\geq 1, \ s\neq 1,2,3$
    $72s+31=3(24s+6)+13$ $24s+6$ $13$ $s\geq 1, \ s\neq 1$
    $72s+37=3(24(s-1)+12)+1$ $24(s-1)+12$ $1$ $s\geq 1, \ s\neq 1$
    $72s+43=3(24s+6)+25$ $24s+6$ $25$ $s\geq 1, \ s\neq 1$
    $72s+49=3(24s+12)+13$ $24s+12$ $13$ $s\geq 1$
    $72s+55=3(24s+18)+1$ $24s+18$ $1$ $s\geq 1$
    $72s+61=3(24s+18)+7$ $24s+18$ $7$ $s\geq 1$
    $72s+67=3(24s+6)+49$ $24s+6$ $49$ $s\geq 1, \ s\neq 1,2,3$
     | Show Table
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