We follow section 1.2 of [CHA].
Here is a brief description of this cryptosystem. Alice is sending a message to Bob. Eve is trying to read Alice's message. Given is a binary code with generator matrix and also given is a decoding algorithm that corrects up to errors. As input parameters are an invertible matrix and an permutation matrix . The triplet will be the secret key and the pair will be the public key. Everyone knows the public key, but Alice and Bob are the only two who know the secret key. For Alice to transmit a bit message , she sends the cipher text , where is a random vector of weight at most . To decipher the cipher text Bob decodes using the given algorithm.
To decrypt an encoded message, consider the matrix