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Let C' be a binary code of length n and distance d 2t +1. Prove that 2"
3. Let C be a q-ary code of length n. Assume the minimal distance d(C) is an odd number, d(C) = 2r + 1. We showed in class that C can always correct up to r errors. That is, whenever a codeword a from C is sent, and r or fewer errors occur in transmission, the Nearest Neighbour Decoding algorithm will decode the received word b correctly (i.e., will decode b as a). Prove that C cannot always correct r...
Let C be a Huffman binary code for source with alphabet S = {s1, · · · , sq}. The code words are w1, w2, · · · , wq. Prove that the equality holds in Kraft’s inequality, i.e., X q k=1 1 2 lk = 1, where lk is the length of wk. Let be a Huffman binary code for source with alphabet S = {sı, . . . , sq). The code words are wi, w2,'.. ,wq. Prove...
If C is a subspace of , prove that . (C is a binary linear code with length n and dimension k, is the dual code of C) F dim(C) dim(C)= n We were unable to transcribe this image
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
Let α,β,γ denote words of length n; d(α,β) denotes the distance between the words α and β. Prove the following triangle inequality: d(α, γ) ≤ d(α, β) + d(β, γ)
Search ll 19:15 1.) (a) binomial relation on N x N Define as (a, b) (c, d)<a + d = b + c Is this binary relation is equivalent relation? If there is an equivalence relation, write three elements of the equivalence class (5,2) to be represented (B)A binary relation on N x N is defined as follows. (a, b)(c, d) a+d<=b + c Will this binary relation be a partial order relation? If it is a partial order relationship,...
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n 2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
(d)n- 1013 2. Let a, b, c, d be integers. Prove the statement or give a counterexample (a) If (ab) c, then a |c and alc. (b) If a l b and c|d, then ac bod (c) If aYb and alc, then aYbc. (d) If a31b4, then alb. (e) If ged(a, b) 1 and alc and b c, then (ab) c. Here a and b are relatively prime integers, also called coprime integers.] rherF and r is an integer with...
I will give a rate! please show work clearly! thanks! 12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A. 12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.