A method performed by a wireless device (510, 800, 1200) for reporting channel state information, CSI, for a downlink channel, the method comprising: estimating a downlink channel; determining, based on the estimated downlink channel, a plurality of coefficients; transmitting (601, 705) a CSI report for the downlink channel to a network node (560, 1100), the CSI report comprising: a quantized representation of a set of reported coefficients; an indication of how the network node is to interpret the set of reported coefficients, wherein the indication of how the network node is to interpret the set of reported coefficients indicates the set of reported coefficients as a subset of a set of candidate reported coefficients; and an indication of a payload size of the set of the reported coefficients, wherein: zero amplitude is not included in a quantization range for the set of reported coefficients; in the quantization range for the set of reported coefficients, the amplitude quantization values are 1 128, 1 / 64, 1 / 32, 1 / 16, 1 / 8, 1 / 4, 1 / 2 and 1; and these amplitude quantization values are associated with the quantization indices 0, 1, 2, 3, 4, 5, 6 and 7, respectively, and wherein the CSI report indicates a plurality of precoder vectors, wherein the precoder vectors are linear combinations of spatial-domain vectors, wherein the coefficients for the spatial-domain vectors in the linear combinations of spatial-domain vectors are parameterized in the frequency-domain as linear combinations of a set of frequency-domain vectors and a set of coefficients for the frequency-domain vectors in the linear combinations of the set of frequency-domain vectors, wherein the reported coefficients are coefficients for the frequency-domain vectors in the linear combinations of the set of frequency-domain vectors, and wherein the frequency-domain vectors are reported in the CSI report, wherein the CSI report comprises a CSI Part 1 and a CSI Part 2. estimating a downlink channel; determining, based on the estimated downlink channel, a plurality of coefficients; transmitting (601, 705) a CSI report for the downlink channel to a network node (560, 1100), the CSI report comprising: a quantized representation of a set of reported coefficients; an indication of how the network node is to interpret the set of reported coefficients, wherein the indication of how the network node is to interpret the set of reported coefficients indicates the set of reported coefficients as a subset of a set of candidate reported coefficients; and an indication of a payload size of the set of the reported coefficients, wherein: a quantized representation of a set of reported coefficients; an indication of how the network node is to interpret the set of reported coefficients, wherein the indication of how the network node is to interpret the set of reported coefficients indicates the set of reported coefficients as a subset of a set of candidate reported coefficients; and an indication of a payload size of the set of the reported coefficients, wherein: zero amplitude is not included in a quantization range for the set of reported coefficients; in the quantization range for the set of reported coefficients, the amplitude quantization values are 1 128, 1 / 64, 1 / 32, 1 / 16, 1 / 8, 1 / 4, 1 / 2 and 1; and these amplitude quantization values are associated with the quantization indices 0, 1, 2, 3, 4, 5, 6 and 7, respectively, and wherein the CSI report indicates a plurality of precoder vectors, wherein the precoder vectors are linear combinations of spatial-domain vectors, wherein the coefficients for the spatial-domain vectors in the linear combinations of spatial-domain vectors are parameterized in the frequency-domain as linear combinations of a set of frequency-domain vectors and a set of coefficients for the frequency-domain vectors in the linear combinations of the set of frequency-domain vectors, wherein the reported coefficients are coefficients for the frequency-domain vectors in the linear combinations of the set of frequency-domain vectors, and wherein the frequency-domain vectors are reported in the CSI report, wherein the CSI report comprises a CSI Part 1 and a CSI Part 2.