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