Information Theory and Coding — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(c) Define a Hamming binary code Ha.in (r, 2.). Let the Parity check matrix of binary Hamming code Decode the received vector 11010`11.'' 3. Parts of the questions to be attempted togetber. I I. (a) Define entropy of a binary source. Calculate the entropy of a source with an alphabet having 64 symbols, where each symbol is equally probable. (7.5) (b) Verify the additive rule of entropies for the probability scheme [A,M = BUC], where the input probabilities are P(A) = 3; P(B) =13; P(C)=15 (c) A transmitter has an alphabet consisting of five letters (x,, X2, Xz, X4, *g) and the receiver has an alphabet of four letters (y, y%, ys= Ya). The joint probabilities are given as : Compute all the entropies for this channel. 6. (a) Construct the standard array for the code having the generator matrix Using this array decode the received vector 01111 (b) Let C be a ternary linear code with generator matrix (i) Find a parity check matrix for C in the standard form. " (i) Use syndrome decoding to decode the received vectors 2121, 1201 and 2222. 5,. (a)Define a generator matrix of a linear code. Prov-e that every- codeworid of a, code is unique.1y ?xpressible as a linear. combination of rows of the gene rator matrix 6f the code. (7.5) (b) .Prove that a code h`aving minimum distance 2t + 1 can correct t or less errors. (7.5) (c).Let C be a linear code over GF(2) having the generator matrix Then find the all code vectors of C. 2, (a).Show that logarithmic.function is convex up.ward function. and. `hence prove tha£ H(X2 S .log(in) where in is the number of source .points. (b) Show that for the joint random variable (X, Y) and the mutual information I(X, Y) (c) Given below the noise characteristic For equi-probable transmission probabilities of the : messages j, Xy, *z, X4 determine the following: (i) Reçeiver entropy H(Y) (i) Conditional entropy H(Y)X) (iii) Mutual in formation I(X;X) (7.5) 3. (a) Define channel capacity, redundancy and efficiency of a noise freé channel. The joint probability matrix of a charinel with binary input and output is given below: Find the different entropies and the mutual information. (7.5) (b) State and prove the Log sum inequality. (c) A codeword of the code {10010; 01010} is transmitted through a BSC with. crossover probability p = 0.2, and a nearest-codeword decoder D is applied to the received word. Compute the decoding error probability Por Of 4. (a) Define A, (n, d) and show thạt A,(5,3) = 4. (b) Listed below are the blocks of a (7, 7, 3, 3, 1) -design. Use this to construct a binary. (7, 16, 3) -code. (c) Show that every square integer is congruent (mod