A binary linear code is an -dimensional subspace of an -dimensional vector space over the binary field . As is an additive abelian group, is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a codeword (element of ) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted received word could have started out as. This procedure is called decoding and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of (a received word could be any element of ) into a standard array. A standard array is a coset decomposition of put into tabular form in a certain way. Namely, the top row of the array consists of the elements of , written in any order, except that the zero vector should be written first. Then, an element of with a minimal number of ones that does not already appear in the top row is selected and the coset of containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of directly above it). This element is called a coset leader and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of containing it is the next row. The process ends when all the vectors of have been sorted into the cosets.
A binary linear code is an -dimensional subspace of an -dimensional vector space over the binary field . As is an additive abelian group, is a subgroup of this group. Codes can be used to correct errors that can occur in transmission. When a codeword (element of ) is transmitted some of its bits may be altered in the process and the task of the receiver is to determine the most likely codeword that the corrupted received word could have started out as. This procedure is called decoding and if only a few errors are made in transmission it can be done effectively with only a very few mistakes. One method used for decoding uses an arrangement of the elements of (a received word could be any element of ) into a standard array. A standard array is a coset decomposition of put into tabular form in a certain way. Namely, the top row of the array consists of the elements of , written in any order, except that the zero vector should be written first. Then, an element of with a minimal number of ones that does not already appear in the top row is selected and the coset of containing this element is written as the second row (namely, the row is formed by taking the sum of this element with each element of directly above it). This element is called a coset leader and there may be some choice in selecting it. Now the process is repeated, a new vector with a minimal number of ones that does not already appear is selected as a new coset leader and the coset of containing it is the next row. The process ends when all the vectors of have been sorted into the cosets.