Meeting the Welch and Karystinos-Pados bounds on DS-CDMA binary signature sets
A \((K,L)\)-signature set \({\mathcal S}=\{s_1,\ldots,s_K\}\) is a set of \(K\) user signatures \(s_i\in\{-1,1\}^L\). The total-squared-correlation (\(TSC\)) is defined by \[ TSC({\mathcal S}) = \sum_{i=1}^K\sum_{j=1}^K | s_is_j^t|^2. \] We denote by \(\theta(K,L)\) the minimum of \(TSC(K<L)\) over all \((K,L)\)-signature sets. A signature set for which the minimum is achieved is called optimal. We have \(TSC({\mathcal S})= \text{trace} ((SS^T)^2)= \text{trace} ((S^TS)^2)\), where \(S\) is the \(K\times L\) matrix with rows the vector \(s_i\). The authors study the question of the existence of optimal signature sets for the following cases when \(K\leq L: L\equiv 1\pmod{4}\), \(K=L\); \(L\equiv 2\pmod{4}\), \(K=L\) or \(K=L-1\). The second case is easy and is settled by Theorem 3. Theorem 3. If \(L\equiv2\pmod{4}\) and there exists a Hadamard matrix of order \(L+2\), then perfect \((K,L)\)-signature sets exist for all \(K\leq L\). For the first case, the authors give various results (Theorems 4--7). Theorem 4. Let \(H\) be a Hadamard matrix of order \(n\). Then \(TSC({\mathcal S}_H)=(n+1)^3 + n(3n+1) - 4\sigma(H)\). Theorem 5. For \(K=L\equiv1\pmod{4}\), if Hadamard matrices of order \(L-1\) exist, then there exists an \((L,L)\)-signature set \({\mathcal S}\) with \(TSC({\mathcal S})=L^3+(L-1)(3L-2)-4\sigma(L-1)\). Theorem 6. Let \({\mathcal C}\) be an \((L,L)\) code with distance distribution \(A_0, A_1, \ldots, A_l\). Then \[ TSC(\phi({\mathcal C}))=L\sum_{i=0}^L A_i(L-2i)^2= L\sum_{j=0}^{(L-1)/2} \bigl(A_{\frac{L-1}{2}-j}+ A_{\frac{L+1}{2}+j}\bigr) (2j+1)^2. \] In particular, \(\phi({\mathcal C})\) is perfect if and only if \(A_i=0\) for \(i\not\in\{0,(L-1)/2,(L+1)/2\}\). Theorem 7. Let \(G=\{g_1,\ldots,g_L\}\) be an Abelian group of order \(L\) and let \(f:G\to\{-1,1\}\) be a binary function on \(G\). Let \({\mathcal S}\) be the \((L,L)\)-signature set \[ {\mathcal S}=\{{\mathbf s}_g=(f(g_1+g), f(g_2+g),\ldots,f(g_L+g))\mid g\in G\}. \] Then \(TSC({\mathcal S}) = L\sum_{h\in G} (C_f(h))^2\), where \(C_f(x)=\sum_{y\in G} f(x+y)f(x)\) is the autocorrelation of \(f\).
scientific article; zbMATH DE number 1975928