In some embodiments, a method includes accepting a code word of a composite Error Correction Code (ECC), which was produced by encoding data with multiple component ECCs, and which was received with one or more reception parameters. One or more of the component ECCs are decoded, but without fully decoding the code word. The one or more reception parameters are estimated based on the decoded component ECCs. In other embodiments, a method includes accepting a code word of an ECC, which encodes data and which was received with one or more reception parameters. An Error Locator Polynomial (ELP), having one or more roots that indicate respective locations of one or more errors in the code word, is derived from the accepted code word. The one or more reception parameters are estimated based on the ELP.