This paper proposes a new class of low-density generator-matrix codes (LDGM) based on chaotic dynamical systems. The codes are designed by controlling symbolic dynamics and using linear convolutional codes. Analyzing the complex structure of chaotic systems, iterative decoding is developed. The communication performance is studied, and convergence analysis of the iterative-decoding system is presented. Finally, comparison and advantages over LDGM linear block codes in terms of encoding complexity and bit-error-rate performance are described, and possible applications of our codes are discussed.