ECE 405/511
Error Control Coding

Assignment Submission

Assignments

1. Consider a Binary Symmetric Channel (BSC) with p = 0.1 (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 = 8 bits
   (b) even parity, word length = 8 bits (7 bits of data)
   
2. Consider a binary code with codewords of length n = 23. This code can correct three errors or less. Calculate the probability of decoding error P(E) and estimate the Bit Error Rate (BER) if this code is used over a BSC with
   (a) p = 0.1
   (b) p = 0.01
   (c) p = 0.001
   (d) p = 0.0001
   
3. Determine the dimension of the binary vector space spanned by the set of vectors
   {(001010),(101000),(001100),(100010),(011111)}
   
4. Find a basis for the dual space of the binary vector space spanned by the set of vectors
   {(11100),(01110),(00111)}
   and determine the vectors in this dual space.
5. Consider the following binary code with 4 codewords
   C = {(00100),(10010),(01001),(11111)}
   (a) What is the minimum distance of this code?
   (b) What is the maximum weight for which the detection of all error patterns is guaranteed?
   (c) What is the maximum weight for which the correction of all error patterns is guaranteed?
   (d) Is this code linear? Provide a proof for your answer.
6. For the code C in Problem 5, find the maximum likelihood codeword associated with the following received vectors
   (a) r = (10110)
   (b) r = (01010)
7. Find the length, dimension and minimum distance for the binary linear code defined by the following parity-check matrix
   H =
   |1010001|
   |1001100|
   |0001011|
   |1101001|
   Also find a generator matrix for this code.

1. Consider the binary code defined by the generator matrix
   G =
   |111000|
   |001110|
   |100101|
   (a) Determine a systematic generator matrix for this code.
   (b) Determine a parity check matrix for the generator matrix obtained in (a).
2. Determine the length, dimension, and minimum distance of the family of extended binary Hamming codes.
3. Write out the codewords in the extended binary Hamming code with n-k=4 and verify the minimum distance obtained in Problem 2. Is this code self-dual? Justify your answer.
4. For the binary linear code with parity check matrix
   H =
   |100011|
   |010101|
   |001111|
   (a) Determine the length, dimension and minimum distance of the code.
   (b) Find a generator matrix for this code.
   (c) Determine a syndrome decoding table for this code and use it to decode the following received vectors
   1. r = 110111
   2. r = 011001
5. Determine using the Hamming bound if it is possible for a binary (9,4,5) code to exist.
6. Does the Gilbert-Varshamov bound verify that a binary (9,2,6) code exists? Can such a code be constructed?
7. The nonbinary Hamming codes have parameters n = (3^q-1)/(q-1), k = n - m, and d_min = 3, for q a prime power. Show that these codes are perfect.

Aaron Gulliver
2026-01-27