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In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the nonsystematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Gro&#x0308;bner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n<sup>3</sup>) using Gaussian elimination to O(qn<sup>2</sup>) with some mild conditions on n and q, where n is the code length and q is the finite-field size.

Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes