Article Contents
Article Contents

# Binary codes from $m$-ary $n$-cubes $Q^m_n$

• * Corresponding author: J. D. Key

The second author is supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071)

• We examine the binary codes from adjacency matrices of the graph with vertices the nodes of the $m$-ary $n$-cube $Q^m_n$ and with adjacency defined by the Lee metric. For $n = 2$ and $m$ odd, we obtain the parameters of the code and its dual, and show the codes to be $LCD$. We also find $s$-PD-sets of size $s+1$ for $s < \frac{m-1}{2}$ for the dual codes, i.e. $[m^2,2m-1,m]_2$ codes, when $n = 2$ and $m\ge 5$ is odd.

Mathematics Subject Classification: Primary: 05C50, 94B05; Secondary: 05B05.

 Citation:

• Table 1.  Blocks in $\mathcal{{B}}$$_m$

 $m$ $5$ $0,2$ $7$ $0,0$ $0,3$ $2,2$ $9$ $0,2$ $0,3$ $2,4$ $11$ $0,0$ $0,3$ $0,4$ $2,2$ $2,5$ $4,4$ $13$ $0,2$ $0,3$ $0,6$ $2,4$ $2,5$ $4,6$ $15$ $0,0$ $0,3$ $0,4$ $0,7$ $2,2$ $2,5$ $2,6$ $4,4$ $4,7$ $6,6$ $17$ $0,2$ $0,3$ $0,6$ $0,7$ $2,4$ $2,5$ $2,8$ $4,6$ $4,7$ $6,8$ $19$ $0,0$ $0,3$ $0,4$ $0,7$ 0, 8 $2,2$ $2,5$ $2,6$ 2, 9 $4,4$ $4,7$ 4, 8 $6,6$ 6, 9 8, 8
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