Aufgaben:Exercise 1.6: (7, 4) Hamming Code

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Code table of the  $\text{HC (7, 4)}$

In 1962,  Richard Wesley Hamming  specified a class of binary block codes that differ in the number  $m$  of check bits supplied. The codeword length is always  $n = 2^m - 1$. The information word consists of  $k = n - m$  bits:

• $m = 2$:   $\text{(3, 1)}$ Hamming code   ⇒   identical to  $\text{RC (3, 1)}$,
• $m = 3$:   $\text{(7, 4)}$ Hamming code,
• $m = 4$:   $\text{(15, 11)}$ Hamming code,
• $m = 5$:   $\text{(31, 26)}$ Hamming code, etc.

In the course of this task there are questions

• to the code size  $ |\hspace{0.05cm}\mathcal{C}\hspace{0.05cm}|$,
• to the code rate  $R$, and
• to the minimum distance  $d_{\rm min}$.

Furthermore, we want to clarify whether the  $\underline{u}_{i} ⇒ \underline{x}_{i}$  given for this task by its code table  $\text{(7, 4)}$-Hamming code is systematic, and whether it is a so-called "perfect code". The running index can take the values  $i = 1, \text{...}\hspace{0.05cm} , 2^k =16$ .

One speaks of a  perfect code if the following condition is satisfied:

$$2^k = \frac{2^n} {\sum_{f=0}^t {n \choose f}}\hspace{0.3cm} \Rightarrow \hspace{0.3cm} 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$

Here  $t$  denotes the number of correctable errors. For odd minimum distance  $d_{\rm min}$  holds:

$$ t = \frac{d_{\rm min}-1 } {2} \hspace{0.05cm}.$$

The interpretation to this condition can be found in the sample solution.

Hints:

• This exercise belongs to the chapter  Examples of Binary Block Codes.
• Reference is made to the chapter  Hamming Codes.

• For this Hamming code, different test equations were used than in the  Theory part. Therefore, the code tables also differ.
• In the  Exercise 1.7, where the same code is used, the chart of the check equations is given.

Questions

1

Specify the characteristics of the given code  $\mathcal{C}$ :

$ \hspace{0.05cm}\mathcal{C}\hspace{0.05cm}| \ = \ $

$ k \hspace{0.46cm} = \ $

$ n \hspace{0.44cm} = \ $

$ R \hspace{0.40cm} = \ $

2

is this a systematic code?

	Yes,
	No.

3

What is the minimum distance between any two code words?

$ \ d_{\rm min} \ = \ $

4

How many transmission errors can be detected  $(e)$  or corrected  $(t)$  respectively?

$e\hspace{0.22cm} = \ $

$t\hspace{0.28cm} = \ $

5

Is the hamming code considered here perfect?

	Yes,
	No.

6

Which statements are true regarding a perfect code?

	A perfect code always results in zero block error probability.
	All receive words  $\underline{y}$  are assignable to a valid codeword.
	For perfect codes, the minimum Hamming–distance is odd.

7

Which of the codes listed below are perfect?

	The  $\text{(15, 11)}$ Hamming code,
	the  $\text{(63, 57)}$ Hamming code,
	the  $\text{(3, 1)}$ repetition code,
	the  $\text{(4, 1)}$ repetition Code,
	the  $\text{(5, 1)}$ repetition code.

Solution

Solution

(1)  The code table has 16 entries: $\underline{|C| = 16}$.

• From the equation $|C| = 2^k$ it follows that $\underline{k = 4}$.
• The length of each codeword is $\underline{n = 7}$.
• Thus, the code rate is $\underline{R = 4/7} = 0.571$.

(2)  Correct YES:

• Each codeword $\underline{x}$ first contains the $k = 4$ bits of the information word $\underline{u}$. This is followed by $m = 3$ check bits:

$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_5,x_7) = ( u_1, u_2, u_3, u_4, p_1, p_2, p_3) \hspace{0.05cm}.$$

• This corresponds exactly to the definition of a systematic code .

(3)  For any Hamming code, the minimum distance $d_{\rm min} \underline{= 3}$.

• From the table, you can see that the minimum Hamming_Weight (the number of ones in a codeword) is equal to 3.
• A linear code, in fact, also includes the zero word, so that holds:

$$d_{\rm min}(\mathcal{C}) = \min_{\substack{\underline{x},\hspace{0.05cm}\underline{x}' \hspace{0.05cm}\in \hspace{0.05cm} \mathcal{C} \ {\underline{x}} \hspace{0.05cm}\ne \hspace{0.05cm} \underline{x}'}}\hspace{0.1cm}d_{\rm H}(\underline{x}, \hspace{0.05cm}\underline{x}') = \min_{\underline{x} \hspace{0.05cm}\in \hspace{0.05cm} \mathcal{C} }\hspace{0.1cm}w_{\rm H}(\underline{x}) = 3 \hspace{0.05cm}.$$

(4)  The specification $d_{\rm min} = 3$ means that $\underline{e = 2}$ errors can be detected and $\underline{t = 1}$ errors can be corrected.

(5)  Correct is YES:

• The condition for a perfect code is according to the specification:

$$ 2^m = {\sum_{f=0}^t {n \choose f}} \hspace{0.05cm}.$$

• In the (7, 4)-Hamming code considered here, $n = 7$, $m = 3$, and $t = 1$, giving a value of 8 on both sides of the equation:

$$ 2^3 = 8\hspace{0.05cm}, \hspace{0.35cm} {\sum_{f=0}^1 {7 \choose f}} = {7 \choose 0} + {7 \choose 1} = 1 + 7 = 8 \hspace{0.05cm}.$$

(6)  Correct statements 2 and 3:

• If there were a channel code with finite codeword length $n$ that made the block error probability zero for all channels, it would not only be perfect, but a miracle.
• But due to the channel coding theorem, ${\rm Pr(block error)} = 0$ is not even possible with finite $n$.

illustration of a perfect code

Let's illustrate statement 2 with a graph. Here, the high-dimensional space is shown in a highly simplified way (in 2D). We assume the numerical values $k = 4$, $n = 7$, $m = 3$ and $t = 1$ of the (7, 4, 3)-Hamming code:

• For the receive word, $2^7 = 128$ points are possible in 7-dimensional space. The red dots mark the $2^4 = 16$ valid codewords.

• The circles include 8 points each, namely a valid codeword and $n = 7$ receive words after only one error, which are assigned to exactly this codeword during decoding.

• Total there are $2^4 = 16$ such circles. Therefore, because of $128 = 16 - 8$, not a single receive word $\underline{y}$ lies outside such an assignment circle.

Also the last statement is true, which shall be shown exemplarily for $d_{\rm min} = 4$:

• Hereby also only $t = 1$ error can be corrected.
• If a receive word $\underline{y}$ differs from permissible code words in two bits, this point is not to be assigned to any circle. Then there are also points outside the circles and the condition of a perfect code is no longer fulfilled.

(7)  Correct are statements 1, 2, 3 and 5:

• All Hamming codes have the minimum Hamming distance $d_{\rm min} = 3$   ⇒   $t = 1$.
• At the same time, any $(n, k)$ Hamming code can also be written as a $(2^m - 1, 2^m - 1 - m)$ code, where $m = n - k$ indicates the number of check bits.
• Thus, the equation of a perfect code is always satisfied:

$${\sum_{f=0}^1 {n \choose f}} = 1 + n = 2^m \hspace{0.05cm}.$$

Where:

• $m = 2$: (3, 1) Hamming code, ⇒ identical to RC (3, 1),
• $m = 3$: (7, 4) Hamming code,
• $m = 4$: (15, 11) Hamming code,
• $m = 5$: (31, 26) Hamming code,
• $m = 6$: (63, 57) Hamming code,

The repetition code with $n = 5$ also satisfies the condition. With $d_{\rm min} = 5$, $t = 2$ and $m = 4$ we obtain:

$${\sum_{f=0}^2 {5 \choose f}} = 1 + 5 + 10 = 16 = 2^m \hspace{0.05cm}.$$

The other repetition codes (RC) with odd $n$ are also perfect, but not with even $n$: RC (4, 1), RC (6, 1), etc. This has already been justified in the sample solution to the subtask (6).