Aufgaben:Exercise 1.08Z: Equivalent Codes

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Four  $(6, 3)$  block codes

In the graph,  the mappings  $\underline{u} \rightarrow \underline{x}$  for different codes are given,  each characterized below by the generator matrix  $\boldsymbol{\rm G}$  and the parity-check matrix  $\boldsymbol{\rm H}$,  respectively:

• ${\boldsymbol{\rm Code \ A}}$:

$${ \boldsymbol{\rm G}}_{\rm A} = \begin{pmatrix} 1 &0 &0 &1 &1 &0\\ 0 &1 &0 &1 &0 &1\\ 0 &0 &1 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},\hspace{0.5cm}{ \boldsymbol{\rm H}}_{\rm A} = \begin{pmatrix} 1 &1 &0 &1 &0 &0\\ 1 &0 &1 &0 &1 &0\\ 0 &1 &1 &0 &0 &1 \end{pmatrix} \hspace{0.05cm}.$$

• ${\boldsymbol{\rm Code \ B}}$:

$${ \boldsymbol{\rm G}}_{\rm B} = \begin{pmatrix} 0 &0 &1 &0 &1 &1\\ 1 &0 &0 &1 &1 &0\\ 0 &1 &1 &1 &1 &0 \end{pmatrix} \hspace{0.05cm},\hspace{0.5cm} { \boldsymbol{\rm H}}_{\rm B} = \begin{pmatrix} 1 &0 &1 &0 &1 &0\\ 1 &1 &0 &1 &0 &0\\ 0 &1 &1 &0 &0 &1 \end{pmatrix} \hspace{0.05cm}.$$

• ${\boldsymbol{\rm Code \ C}}$:

$${ \boldsymbol{\rm G}}_{\rm C} = \begin{pmatrix} 1 &0 &0 &1 &0 &1\\ 0 &1 &0 &0 &1 &1\\ 0 &0 &1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm},\hspace{0.5cm}{ \boldsymbol{\rm H}}_{\rm C} = \begin{pmatrix} 1 &0 &1 &1 &0 &0\\ 0 &1 &1 &0 &1 &0\\ 1 &1 &1 &0 &0 &1 \end{pmatrix} \hspace{0.05cm},$$

• ${\boldsymbol{\rm Code \ D}}$:

$${ \boldsymbol{\rm G}}_{\rm D} = \begin{pmatrix} 1 &0 &0 &1 &0 &1\\ 0 &1 &0 &1 &0 &0\\ 0 &0 &1 &0 &1 &0 \end{pmatrix} \hspace{0.05cm},\hspace{0.5cm}{ \boldsymbol{\rm H}}_{\rm D} = \begin{pmatrix} 1 &1 &0 &1 &0 &0\\ 0 &0 &1 &0 &1 &0\\ 1 &0 &0 &0 &0 &1 \end{pmatrix} \hspace{0.05cm}.$$

This task is to investigate which of these codes or code pairs are

• are systematic,
• are identical  (that is:   Different codes have same code words),
• are equivalent  (that is:   Different codes have same code parameters).

Hints :

• This exercise belongs to the chapter  "General Description of Linear Block Codes".

• Reference is made in particular to the pages  "Systematic Codes"   and   "Identical Codes".

• Note that the specification of a parity-check matrix  $\boldsymbol{\rm H}$  is not unique.  If one changes the order of the parity-check equations, this corresponds to a swapping of rows.

Questions

1

Which of the codes listed below are systematic?

	Code  $\rm A$,
	Code  $\rm B$,
	Code  $\rm C$,
	Code  $\rm D$.

2

Which of the given code pairs are identical?

	Code  $\rm A$  and  code  $\rm B$,
	Code  $\rm B$  and  code  $\rm C$,
	Code  $\rm C$  and  code  $\rm D$.

3

Which of the given code pairs are equivalent but not identical?

	Code  $\rm A$  and  code  $\rm B$,
	Code  $\rm B$  and  code  $\rm C$,
	Code  $\rm C$  and  code  $\rm D$.

4

How do the generator matrices  $G_{\rm B}$  and  $G_{\rm C}$  differ?

	By different linear combinations of different rows.
	By cyclic shifting of rows by  $1$  down.
	By cyclic shifting of columns by  $1$  to the right?

5

For which codes applies  ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$?

	Code  $\rm A$,
	Code  $\rm B$,
	Code  $\rm C$,
	Code  $\rm D$.

Solution

Solution

(1)  Correct are the answers 1, 3 and 4:

• For a systematic (6, 3) block code, the following must hold:

$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_6) = ( u_1, u_2, u_3, p_1, p_2, p_{3}) \hspace{0.05cm}.$$

This condition is satisfied by code A, code C, and code D, but not by code B.

(2)  Correct is only answer 1:

• Only code A and code B are identical codes. They contain exactly the same code words and differ only by other assignments $\underline{u} \rightarrow \underline{x}$.
• As indicated in the sample solution to Exercise 1.8 (3), one gets from the generator matrix ${ \boldsymbol{\rm G}}_{\rm B}$ to the generator matrix ${ \boldsymbol{\rm G}}_{\rm A}$

• by swapping/permuting rows alone, or
• by replacing a row with the linear combination between that row and another.

(3)  Thus, the correct answer is answer 2 alone:

• Code A and code B are more than equivalent, namely identical.
• Code C and D also differ, for example, by the minimum Hamming distance $d_{\rm min} = 3$ and $d_{\rm min} = 2$, respectively, and are thus also not equivalent.
• Code B and code C, on the other hand, show the same properties, for example $d_{\rm min} = 3$ holds for both. However, they contain different codewords.

(4)  Correct is answer 3:

• The last column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the first column of ${ \boldsymbol{\rm G}}_{\rm C}$.
• The first column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the second column of ${ \boldsymbol{\rm G}}_{\rm C}$.
• The second column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the third column of ${ \boldsymbol{\rm G}}_{\rm C}$, etc.

(5)  All statements are true:

• The condition ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$ holds for all linear codes.