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This paper addresses the use of multirate filter banks in the context of error-correction coding. An in-depth study of these filter banks is presented, motivated by earlier results and applications based on the filter bank representation of Reed-Solomon (RS) codes, such as Soft-In Soft-Out RS-decoding or RS-OFDM. The specific structure of the filter banks (critical subsampling) is an important aspect in these applications. The goal of the paper is twofold. First, the filter bank representation of RS codes is now explained based on polynomial descriptions. This approach allows us to gain new insight in the correspondence between RS codes and filter banks. More specifically, it allows us to show that the inherent periodically time-varying character of a critically subsampled filter bank matches remarkably well with the cyclic properties of RS codes. Secondly, an extension of these techniques toward the more general class of BCH codes is presented. It is demonstrated that a BCH code can be decomposed into a sum of critically subsampled filter banks.