Optimal Encodings of Linear Block Codes for Unequal Error Protection

It is possible for a linear block code to provide more protection for selected message positions than is guaranteed by the minimum distance of the code. The protection provided a message position can be measured by associating a number with that position called its separation. The separation of a message position measures the protection provided to that position in a manner analogous to that in which the minimum distance of a code measures the protection provided the entire message. This paper proves that any fixed linear block code has an encoding which is optimal with respect to the error protection provided the individual message positions. More precisely, among those encodings of the code for which the separations associated with the message positions are arranged in nondecreasing order, there is at least one which simultaneously maximizes all the separations associated with the message positions. A procedure is given which may be used to construct optimal encodings for linear codes of small dimension. When the Hamming metric is employed, the procedure builds a generator matrix which is as sparse as possible for the given code. At each iteration the procedure adds a row to a partially constructed generator matrix. A code word of minimum weight is chosen for this purpose—subject to the restriction that the rows of the generator matrix must be linearly independent. A more general result is that any generator matrix which is as sparse as possible induces an optimal encoding of its row space. A similar result holds when the Lee metric is used to model a channel. Theorems dealing with cyclic codes and product codes are developed. Under suitable restrictions, an optimal generator matrix for a cyclic code may be formed by concatenating the generator matrices of the minimal ideals which are contained in it. When the Hamming metric is employed, an optimal generator matrix for a product code may be obtained by taking the Kronecker product of optimal generator matrices for the component codes.