Chapter 5, Binary Linear Codes Video Solutions, Foundations of coding: theory and applications of error-correcting codes

Problem 1

Is the binary code of all palindromes (i.e., words which are the same when read backward or forward) linear? Describe it by equations, and determine the number of errors it detects.

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Problem 2

Describe an algorithm for detection of triple errors when the rectangular code (5.2) is used.

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Let $K$ be the binary code of all words of length 7 such that (a) the third bit is a parity check for the first two bits, (b) the sixth bit is a parity check for the fourth and fifth bits, and (c) the last bit is an overall parity check. Describe $K$ by equations and determine the number of errors it can correct or detect.

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03:17

Problem 4

The rectangular code (5.2) can also be called a two-dimensional evenparity code. Define, more generally, $m$-dimensional even-parity codes. For which $m$ is such a code capable of correcting double errors? And triple errors?

Stanley Enemuo

Numerade Educator

Problem 5

Hamming codes:
(1) Compute the error probability $P_{\text {err }}(K)(4.8)$ of the Hamming code $K$ of length 7 used in a binary symmetric channel corrupting 1 bit per hundred on the average. (Hint: use the error estimate in 4A.)
(2) More in general, express the error probability of the Hamming code of length $2^m-1$ as a function of $m$.
(3) What is the probability $R_{\mathrm{und}}(K)$, of an undetected errror of the Hamming code of length $2^m-1$ ?

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03:17

Problem 6

Prove that every single-error-correcting code of length $2^m-1$ has at least $m$ check symbols.

Stanley Enemuo

Numerade Educator

Problem 7

Prove that the decoding of Hamming codes is always incorrect if two bits are corrupted in a code word.

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Problem 8

A binary linear code of length 8 is described by the following equar tions:
$$
\begin{aligned}
& x_5=x_2+x_3+x_4, \\
& x_6=x_1+x_2+x_3, \\
& x_7=x_1+x_2+x_4, \\
& x_8=x_1+x_3+x_4 .
\end{aligned}
$$

Find its parity check matrix and verify that the minimum distance is 4 .

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Foundations of coding: theory and applications of error-correcting codes

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11