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We present the design and analysis of a family of codes based on the efflciently-encodable rate-compatible irregular LDPC codes (E2RC codes) introduced by Kim, Ramamoorthy and Mclaughlin '06. The basic idea is to utilize the structured parity part (of the parity check matrix) of E2RC codes and design optimal degree distributions for the systematic part based on EXIT chart analysis. In this work, we propose a new technique for computing EXIT functions of the structured parity part without resorting to Monte Carlo simulations. Our method provides smoother EXIT functions in substantially lesser time. Furthermore, we pose and solve the problem of designing good codes using linear programming. Next, we consider the performance of rate-compatible codes under puncturing using EXIT charts. Finally, we propose joint optimization of our codes at any specified code rate(s) which has not been explored in previous design of rate-compatible punctured codes.