Systems and methods provide an optionally keyed error-correcting code that is spectrally concentrated. Each codeword of the low dimensional spectral concentration code (LDSC code) typically has very few coefficients of large magnitude and can be constructed even with limited processing resources. Decoding can be performed on low power devices. Error-correcting code is constructed around a key using basic computer arithmetic for computations instead of finite field arithmetic, thus saving energy. A recipient who possesses the key enjoys correction of a relatively high percentage of noise errors. In one implementation, a direct list-decoder iteratively estimates a list of message words directly, instead of a list of codewords. In variations, a unique message word is selected from the list either by applying a randomness test or by using message passing.