Download Problem Set 5 for ENEE626 (2008) - Constructing and Decoding Product Codes - Prof. Alexand and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! ENEE626 (2008). Problem set 5. Due in class on Dec. 9. Let C = C1 ⊗ C2 be a product code, where C1[n1, k1, d1], C2[n2, k2, d2] are the column and the row code, respectively. (a) (10pt) In class we proved that the distance d(C) ≥ d1, d2. Show explicitly how to construct a code- word of the code C that has weight d1, d2 (so the inequality is in fact an equality). (b) (10pt) Let H1 and H2 be parity-check matrices of the component codes. Write out a parity-check matrix of the code C . (c) (10pt) Let C1 = C2 = [7, 4, 3] binary linear simplex code. Take its generator matrix in the form H = [ 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 ] Coordinates 2, 3, 4 form an information set (why?). Using these coordinates as message positions and as- suming systematic encoding, write out the codeword that corresponds to the message [ 0 1 0 1 1 0 0 1 1 ] in the product code C = C1 ⊗ C2. Show the locations of the message symbols in the obtained codeword. (d) (10pt) Decode the received word y = 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 with the code C using the algorithm that corrects (d(C ) − 1)/4 errors. (e) (15pt) Decode the vector y in part (d) using the min-sum algorithm, showing all the calculations performed by it. Remark. Observe that even the simple algorithm in part (d) will correct many combinations of 6,7 errors. Thus, to force it into making an error, the error pattern must be of a special form. (f) (10pt) Consider a complete bipartite graph G(V = V1 ∪ V2, E), i.e., a graph with the property that every two vertices v ∈ V1, w ∈ V2 are connected with an edge, and there are no edges between vertices v,w if both v,w ∈ V1 or if both v,w ∈ V2. Let |V1| = n2, |V2| = n1. Consider a binary code D whose coordinates are in one-to-one correspondence with the set E. For a codeword c ∈ D let c(v) be the subvector of c indexed by the edges that are incident to v. By definition, c ∈ D ⇔ ∀v∈V1c(v) ∈ C1 & ∀w∈V2c(w) ∈ C2. Prove that D = C1 ⊗ C2 (refer to the example in part (e)). (e) (5pt) Assume that the received vector is y as above, that the vertices in V1 (V2) correspond to the columns (resp., to the rows) of the matrix y, and that the edges are numbered so that they match the figure on the next page. Show the steps of the min-sum decoding algorithm performed on the graph by specifying which information is sent from the left vertices to their neighbors along the edges, and which calculation is performed by each “message” vertex on the right.
