The spectrum \(Q(k,\lambda)\) of coset difference arrays with \(k=2\lambda+1\) (Q2747187)

The authors study the spectrum \(Q(k,\lambda)\) of coset difference arrays and determine the special case \(Q(3,1)\). For \(q,k,\lambda\) integers, \(q\) prime power, \(\delta = {k-1 \over {\text{gcd}}(k-1,\lambda)}\), and \(q-1\) divisible by \(k\delta\), a \((q,k,\lambda)\)-coset difference array is defined as follows. Let \(H\) be the subgroup of order \({q-1\over k\delta}\) of the multiplicative group \(\text{GF}(q)^*\), and \(H_0,\ldots,H_{k\delta -1}\) be its cosets. Then the coset difference array is a \(\delta\times k\)-matrix \(b_{ij}\) of elements of \(\text{GF}(q)^*\) with the properties (1) \(b_{ij}\in H_{j\delta +i}\) for all \(i=0,\ldots,\delta -1\) and \(j=0,\ldots,k-1\), (2) for fixed \(j, j^\prime\) all differences \(b_{ij} - b_{i{j^\prime}}\) belong to the same coset, and (3) each coset contains the same number of these differences. The spectrum \(Q(k,\lambda)\) is the set of all prime powers \(q\) for which a \((q,k,\lambda)\)-coset difference array exists. The authors give for \(k=2\lambda-1\) a bound from which on all relevant prime powers \(q\) belong to the spectrum.