[math-fun] *linear* decoding of ECC code ?

The simplest form of ECC is that for a single bit: 0 -> 000, 1 -> 111. The decoding rule is also a simple majority test: 001/010/100 -> 0; 110/101/011 -> 1. My problem: I'd like an ECC code that can be computed *linearly* -- i.e., via a (chain of) matrix multiplication(s). Thus, codeword . R . P = decodedword Where R is an invertible matrix and P is some non-invertible matrix -- e.g., a projection matrix. I haven't been able to find a linear decoding for this version of ECC, nor have I been able to convince myself that it is impossible. If this type of ECC can't be decoded in a linear fashion, is there any other type of ECC that can?