Let M be a cyclic incidence matrix for a projective plane with S lines and points. Thus each row has n 1s. The largest eigenvalue is n, and all remaining eigenvalues have modulus = n - 1. A bipartite graph consisting of two sets of S nodes of order n is described by the connection matrix 0 M A = Mt This matrix may be seen as a simple expander graph: Starting from a node in the right set, n nodes in the left set can be reached in one transition, and the remaining nodes in the right set can be reached from these nodes. We have S = n(n - 1) + The graph can be used to define a code by associating a symbol with each branch and letting all branches that meet in a node satisfy the parity checks of an (n, k, d) code. Thus the length of the code is N = Sn If the rate of the code associated with the nodes is r, the total rate is R 2r - For our purpose we will choose codes on both sides of the graph as extended RS codes over the field of q = n - 1 symbols, since in this way the same field is used in constructing the projective plane. But we will refer to the codes on the right as the inner codes, since the symbols are converted to binary vectors, and the right side is decoded as binary codes.
In order to obtain a lower bound on the minimum distance, we want to determine the smallest size of sets of nodes in each of the two parts of the graph, s, such that the subgraph consisting of these nodes and the branches connecting them has degree at least d. Clearly in this case sd is a lower bound on the minimum distance. It follows from the expansion property of the graph that the minimum distance satisfies the product bound if d >> n.
The decoding can make use of the graph structure of the code. First decode the binary images of the right side codes. For each F (q) symbol in a given position, propagate a message along the branch in the graph indicating the minimum number of binary errors corrected in the first stage of decoding. Using these messages, decode the left codes as RS codes. Pass the result to the right side, and consider these codes as RS codes. Each code on the right side is now the root of a tree code consisting of all codes on the right side, a small subset of codes on the left side, and all symbols in the total code. To get a complete decoding, the results from several of these trees must be reconciled.
8. CONCLUDING REMARKS This presentation mentions only a small part of the research on concatenated codes. The topics discussed here reflect my personal interests, and they were chosen to illustrate some of the interactions between coding ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 4 № 3 PROBLEMS OF CONCATENATED CODES theory and related subjects and between different areas of coding theory. I consider it a privilege to have been involved with this line of research, both on the scientific side and in applications over many years.
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