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On the combinatorial structure of a class of \(\left[ \binom m 2, \binom{m-1}{2}, 3\right]\) shortened Hamming codes and their dual-codes. (English) Zbl 1184.05105
Summary: Let \(\mathcal H\) be the binary linear block code with parity-check matrix \(H_m\) whose columns are all distinct binary strings of length \(m\) and Hamming weight 2. It is shown that \(\mathcal H\) is an \([n,d,k]=\left[\frac{m(m-1)}{2},\frac{(m-1)(m-2)}{2},3\right]\) code while the dual-code \({\mathcal H}^\perp_m\) has dimension \(k^\perp\) and minimum distance \(d^\perp\) satisfying \(k^\perp=d^\perp=m-1\). It is in general very difficult to find or even estimate the covering radius of a given code. It is shown here that the covering radius of \({\mathcal H}_m\), denoted \(\text{Cr}({\mathcal H}_m)\), is \(\lceil \frac m2\rceil\). We also show that \(\text{Cr}({\mathcal H}^\perp_m)=\frac{m(m-2)}{4}\) if \(m\) is even and \(\text{Cr}({\mathcal H}^\perp_m)=\frac{(m-1)^2}{4}\) if \(m\) is odd. Thus \(\text{Cr}({\mathcal H}^\perp_m)\simeq\text{Cr}({\mathcal H}_m)^2\). The weight distribution of \({\mathcal H}^\perp_m\) is given. This together with the MacWilliams identities results in an expression for the weight distribution of \({\mathcal H}_m\). It turns out that the covering radius of \({\mathcal H}_m\) is equal to its external distance. From the Tanner graph perspective, the Tanner graphs of \({\mathcal H}_m\) and \({\mathcal H}^\perp_m\) have girth 6. It is shown that the Tanner graphs of \({\mathcal H}^\perp_{m+1}\) and \({\mathcal H}_m\) are essentially identical and are structurally representable by the complete graph \(K_m\) on \(m\) vertices.
MSC:
05C75 Structural characterization of families of graphs
05C90 Applications of graph theory
94B05 Linear codes, general
Software:
LDGM
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