ECE 405/511
Error Control Coding

Assignment Submission

Assignments

1. Consider a Binary Symmetric Channel (BSC) with p = 0.01 (the probability that a bit is received in error). Compute the probability that a received word contains undetected errors given the following coding schemes
   (a) no coding, word length = 5 bits
   (b) even parity, word length = 6 bits (5 bits of data)
   
2. Consider a binary code with codewords of length n = 23 and dimension k = 12. This code can correct three errors or less. It is used with (nonconherent) BFSK modulation over an AWGN channel with SNR = 13 dB.
   (a) Calculate the probability of error p without coding.
   (b) Calculate the probability of decoding error P(E) and estimate the Bit Error Rate (BER).
   (c) What is the coding gain in dB?
   
3. Find a basis for the dual space of the binary vector subspace spanned by the set of vectors
   {(10011),(11100),(00111)}
   and determine the vectors in this dual space.
4. Determine the dimension of the binary vector subspace spanned by the set of vectors
   {(01111),(11001),(01110),(11000)}
   
5. Consider the following binary code with 4 codewords
   C = {(011000),(100100),(010111),(101011)}
   (a) Is this code linear? Provide a proof for your answer.
   (b) What is the minimum distance of this code?
   (c) What is the maximum weight for which the detection of all error patterns is guaranteed?
   (d) What is the maximum weight for which the correction of all error patterns is guaranteed?
6. For the code C in Problem 5, find the maximum likelihood codeword associated with the following received vectors
   (a) r = (100000)
   (b) r = (011111)
   (c) r = (111100)
7. Find the length, dimension, rate, and minimum distance for the binary linear code with the following generator matrix
   G =
   |111000|
   |001110|
   |100101|
   Determine a systematic generator matrix for this code and give a parity check matrix for this systematic generator matrix.

Aaron Gulliver
2025-01-17