In code-based public key encryption schemes, a public key is formed by matrix-multiplying 2 linear matrices to the left and right side of a easily decodeable error-correcting code, so that it'll be difficult to extract useful information that may be used to decrypt ciphertexts.
In multivariate digital signature schemes, a public key is formed by compositing linear equation systems to the inner and outter parts of a easily solvable multivariate (usually quadratic) equation system, so that the resulting composition cannot be easily reversed.
From my understanding, the composition with linear systems is the biggest similarity between code-based PKE and multivariate DSS, and I wonder:
Is there any other aspect where code-based and multivariate cryptosystems are similar?
What are the important differences between code-based and multivariate cryptosystems?