Download Homework Solutions for ENEE739C: Coding Theory - Prof. Alexander Barg and more Assignments Electrical and Electronics Engineering in PDF only on Docsity! ENEE739C: Homework 1. Date due 10/23/2003. (no e-mails please). 1. We have proved (Thm. 2.6) that there exists an [n, k] binary linear code whose weight distribution A1, A2, . . . , An is bounded above as Aw ≤ n2 ( n w ) 2k−n if the right-hand side is at least one (and Aw = 0 otherwise). Prove that it is possible to improve this estimate by replacing n2 with n : there exists a binary linear [n, k] code with Aw ≤ n ( n w ) 2k−n. Hint: consider the ensemble of all linear [n, k] codes. Find the probability that a nonzero vector is contained in a random code from the ensemble. Compute the expected number of weight-w vectors, use the Markov inequality. 2. Consider a finite metric space X in which the volume of the ball Br(x) depends on its center. Let 〈Br〉 = 1 |X| ∑ x∈X vol(Br(x)) be the average volume of the ball of radius r. Prove that X contains a code of minimum distance d and size M ≥ |X|/(4〈Bd−1〉). Hint: perform the Gilbert procedure on the subset of all the points y ∈ X such that vol(Bd−1(y)) ≤ 2〈Br〉. 3. (GV bound in the Johnson space) Let J n,w be the space of all binary vectors of length n and weight w. (a) Let x,y ∈J n,w. Prove that the Hamming distance d(x,y) is even. (b) Let C be a code of rate R and minimum distance d = 2δn. Prove that J n,w contains codes of rate approaching the bound R = h2(ω)− ωh2 ( δ ω ) − (1− ω)h2 ( δ 1− ω ) . 4. Recall the inequality used to prove Thm. 5.1 (1) Pe(C) ≤ min t [Pe(C,y ∈ Bt(0)) + P (y 6∈ Bt(0))]. where y is the the error vector. In the proof we took C a random linear code and chose t = dGV. Prove that for large n this choice is optimal, i.e., furnishes the minimum on t in (1). Hint: Find the smallest t satisfying, for large n, 2−n(1−R) 2t∑ w=d ( n w ) t∑ r=dw/2e [ w∑ i=w/2 ( w i )( n− w r − i )] pr(1− p)n−r ≥ n∑ r=t+1 ( n r ) pr(1− p)n−r. 5. Consider an [n = 8, k, d] linear code C given by the row space of the matrix G = 1 1 1 1 1 1 1 10 0 0 0 1 1 1 10 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 Find k, d. Write out a parity-check matrix of C. Let E = {1, 2, 3}. Write out the parameters of the codes CE , CE , (C⊥)E , (C⊥)E . Explain all answers. 6. Let C[n, k, d] be a linear q-ary MDS code, i.e., a code with d = n− k+ 1. Prove that the dual code C⊥ is also MDS. 1
