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E2 205 (Aug - Dec 2021 Online Semester) : Homework Assignment 8

This document provides the instructions for Homework Assignment 8 for the online E2 205 course. It includes 3 problems related to error correcting codes. Problem 1 involves using the Peterson-Gorenstein-Zierler decoding algorithm to decode 3 received words of a (15,5,7) BCH code. Problem 2 analyzes the extended Euclidean algorithm decoder for syndromes where certain elements are fixed. Problem 3 involves finding minimum weight codewords corresponding to the syndromes in Problems 1a and 1b. Students are allowed to collaborate but must list their collaborators. Problems marked with an asterisk must be solved independently.

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Kurada Ravindra
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31 views1 page

E2 205 (Aug - Dec 2021 Online Semester) : Homework Assignment 8

This document provides the instructions for Homework Assignment 8 for the online E2 205 course. It includes 3 problems related to error correcting codes. Problem 1 involves using the Peterson-Gorenstein-Zierler decoding algorithm to decode 3 received words of a (15,5,7) BCH code. Problem 2 analyzes the extended Euclidean algorithm decoder for syndromes where certain elements are fixed. Problem 3 involves finding minimum weight codewords corresponding to the syndromes in Problems 1a and 1b. Students are allowed to collaborate but must list their collaborators. Problems marked with an asterisk must be solved independently.

Uploaded by

Kurada Ravindra
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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E2 205 (Aug – Dec 2021 online semester)

Homework Assignment 8
Submission deadline: Thursday, Oct. 21, 11:59pm

Questions marked with ∗ will not be discussed in the tutorial sessions. You have to solve them on your own.

You are allowed to collaborate with your classmates in solving the homework problems, including on the ∗ questions,
but you must then write down the names of your collaborators for each problem.

∗ 1. Let C be the [15, 5, 7] triple-error-correcting binary BCH code with parity-check matrix
 
1 α α2 α3 ··· α14
2 2 2 3 2 2
1 α (α ) (α ) · · · (α14 )
 
 
HRS = .. .. .. .. .. .. ,
. . . . . .
 
 
6 2 6 3 6 6
1 α (α ) (α ) · · · (α14 )

where α ∈ F16 is a root of f (x) = x4 + x3 + 1. Suppose that a codeword from C was transmitted. Use the
Peterson-Gorenstein-Zierler decoding algorithm to decode the following received words, if possible:

(a) 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
(b) 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0
(c) 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

2. Let C be the code in Problem 1 above. Let s = [s0 s1 s2 s3 s4 s5 ]T ∈ (F16 )6 be the syndrome arising from
a binary received word y ∈ (F2 )15 . In each of the following cases, analyze the operation of the extended
Euclidean algorithm decoder to identify all the values of s4 or s2 (as the case may be) for which the decoder
is able to produce a codeword from C as output.

(a) s0 = 0, s2 6= 0, s4 = 0.
(b) s0 = 0, s2 = 0, s4 6= 0.

3. (a) The received word in Problem 1(a) yields a syndrome of the form given in Problem 2(a). Find a binary
word y of least Hamming weight that gives rise to a syndrome of this form.
(b) The received word in Problem 1(b) yields a syndrome of the form given in Problem 2(b). Find a binary
word y of least Hamming weight that gives rise to a syndrome of this form.

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