When a codeword c is sent and a vector r is received, the difference is known as **the error vector e**, i.e., r = c + e. If H is a parity-check matrix for the linear code C, HrT = H (c + e) T = HcT + HeT = HeT, because HcT = 0 for every codeword. The syndrome of r is a term used to describe HrT. The syndrome can also be defined as the column vector obtained by parsing out the erasures in r from its equivalent vector in GF(2). The number of nonzero entries in this vector is called the syndrome length and is denoted by k.

Syndromes. The syndrome of x is defined as s = Hx for every (row) vector x in the ambient vector space. If and only if s = 0, the vector x is a codeword. The technique for decoding syndromes is based on the computation of syndromes. Decoding can also be done by finding the vector which has **the minimum-weight zero-syndrome vector**.

Syndrome decoding is a highly efficient technique of decoding **a linear code** over a noisy, error-prone channel. Syndrome decoding is essentially minimal distance decoding with **a smaller lookup table**. The linearity of the code allows for this. If one assumes that errors occur independently and identically distributed (iid) then it can be shown that the probability of incorrectly decoding a codeword is equal to **its noise rate**.

A syndrome is a number which tells you whether or not a bit is wrong. For example, if your binary message string was 10101010, then the syndrome would be 0 1 0 1. Since there is an even number of ones in the message string, the decoder knows that none of the bits are bad and can be set to zero.

If the number of ones in the syndrome is odd, then at least one bit is incorrect. The decoder will then need to choose between two options: try another value for the bad bit, or stop decoding and report an error. In our example, since there are three ones in the syndrome, we know that at least one bit is incorrect and should be replaced. We can either choose to keep trying until we find a good value or report an error. In fact, under these circumstances, there is only a 50% chance of correctly decoding the message string.

The identical condition is referred to as a "coset" of the code C. An equivalence relation is having the same syndrome, and the cosets are the equivalence classes of this equivalence relation. For some x Zn 2, 2 takes the form x + C. Each coset, in particular, has |C| = 2m elements, and there are 2n-m cosets. If n is even, then there are m cosets with even numbers of elements and m cosets with odd numbers of elements.

In mathematics, a coset is the set of all elements in a group that can be written as a single element multiplied by **some other element** in the group. In the context of linear algebra, the term coset refers to any one of the nonzero solutions to a system of **linear equations** with integer or rational coefficients. In this case, the solution set forms a group under vector addition, with each solution acting as an identity element for multiplication. The number of solutions to a linear equation over the integers (or more generally, to some ring) is often called the dimension of the corresponding linear space. A basis for such a space can be used to write any other vector in the space in terms of these solutions; thus, it allows one to reduce the problem of finding all solutions to such an equation to that of finding those vectors whose components are zero except for ones in a basis. Linear spaces with integer or rational bases are called integral or rational linear spaces, respectively.