Aufgaben:Exercise 2.07: Reed-Solomon Code (7, 3, 5) to Base 8

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$\rm GF(2^3)$ representation as powers, polynomials, vectors

The Reed–Solomon code considered here,  denoted  $\rm RSC \, (7, \, 3, \, 5)_8$,

• encodes a block of information  $\underline{u} = (u_0, \, u_1, \, u_2)$  of  $k = 3$  symbols,  where  $u_0, \, u_1, \, u_2 ∈ \rm GF(2^3)$  applies,  representing symbols rather than bits;

• generates a code word   $\underline{c} = (c_0, \, c_1,\hspace{0.05cm} \text{...} \hspace{0.1cm} , c_6)$   of length  $n = 7$  with code symbols  $c_i$  also from the Galois field  $\rm GF(2^3)$;

• has the free distance   $d_{\rm min} = n - k + 1 = 5$,  so that up to   $e = 4$   symbol errors can be detected and up to   $t = 2$   symbol errors can be corrected.

The elements of the underlying Galois field are:

$${\rm GF}(2^3) = \{\hspace{0.1cm} 0\hspace{0.05cm},\hspace{0.1cm} 1 \hspace{0.05cm},\hspace{0.1cm} \alpha\hspace{0.05cm},\hspace{0.1cm} \alpha^{2} \hspace{0.05cm},\hspace{0.1cm} \alpha^{3} \hspace{0.05cm},\hspace{0.1cm} \alpha^{4} \hspace{0.05cm},\hspace{0.1cm} \alpha^{5} \hspace{0.05cm},\hspace{0.1cm} \alpha^{6} \hspace{0.1cm}\}\hspace{0.05cm}. $$

These elements can also be represented as polynomials or as coefficient vectors according to the graphic.

1. One can see from the above table that all  $u_i ∈ \rm GF(2^3)$  and all  $c_i ∈ \rm GF(2^3)$  can also be characterized by  $m = 3 \ \ \rm bits$,  for example  $\alpha^4$  by  "$110$".
2. In this exercise,  you are to trace the encoding process for the binary input sequence

$$110 \hspace{0.09cm}001\hspace{0.09cm} 011\hspace{0.09cm} 000\hspace{0.09cm} 000\hspace{0.09cm} 000\hspace{0.09cm} 111 \hspace{0.05cm} \text{...} $$

Note that:

• The Reed–Solomon coder works blockwise.  In the first coding step,  the code symbols   $c_0, \hspace{0.05cm} \text{...} \hspace{0.1cm} , c_6$   are generated,  then in the second step from the information block  $\underline{u} = (u_3, \, u_4, \, u_5)$  the symbols   $(c_7, \hspace{0.05cm} \text{...} \hspace{0.1cm} , c_{13})$   of the second code word, etc.
• One describes the information block  $\underline{u}$  by the polynomial   $u(x) = u_0 + u_1 \cdot x + u_2 \cdot x^2$   of degree  $2$.  In general,  for the Galois field  $\rm GF(2^m)$  the degree of the polynomial results to  $m - 1$.
• The code symbols   $c_0, \hspace{0.05cm} \text{...} \hspace{0.1cm} , c_6$   are obtained by substituting into the polynomial  $u(x)$  for the parameter  $x$  all elements of  $\rm GF(2^3)$  except the zero element:

$${\rm GF}(2^3)\hspace{-0.01cm}\setminus \hspace{-0.01cm}\{ 0\} = \{\hspace{0.1cm}\alpha^{0}\hspace{0.05cm},\hspace{0.1cm} \alpha^1\hspace{0.05cm},\hspace{0.1cm} \alpha^{2} \hspace{0.05cm},\hspace{0.1cm} \alpha^{3} \hspace{0.05cm},\hspace{0.1cm} \alpha^{4} \hspace{0.05cm},\hspace{0.1cm} \alpha^{5} \hspace{0.05cm},\hspace{0.1cm} \alpha^{6} \hspace{0.1cm}\}\hspace{0.05cm}. $$

• Formally, the  $\rm RSC \, (7, \, 3, \, 5)_8$  can be described as follows:

$$C_{\rm RS} = \Big \{ \underline {c} = \big ( u(\alpha^{0}) \hspace{0.05cm},\hspace{0.1cm} u(\alpha^{1})\hspace{0.05cm},\hspace{0.1cm}u(\alpha^{2})\hspace{0.1cm} \big ) \hspace{0.1cm} \big | \hspace{0.2cm} u(x) = \sum_{i = 0}^{2} u_i \cdot x^{i}\hspace{0.05cm},\hspace{0.2cm} u_i \in {\rm GF}(2^3) \Big \} \hspace{0.05cm}.$$

Hints:

• This exercise belongs to the chapter  "Definition and Properties of Reed–Solomon Codes".

• The  "Exercise 2.8"  is structured similarly to this one,  but the generator matrix  $\mathbf{G}$  is then used to generate a code word  $\underline{c}$.

Questions

1

What is the binary information block in the first coding step?

	$\underline{u}_{\rm bin} = (110)$,
	$\underline{u}_{\rm bin} = (110_|001_|011)$,
	$\underline{u}_{\rm bin} = (110_|001_|0)$.

2

What are the information symbols in the first coding step?

	$u_0 = \alpha^4$,
	$u_0 = 0$,
	$u_1 = \alpha^6$,
	$u_1 = \alpha^0$,
	$u_2 = \alpha^3$,
	$u_2 = \alpha^2$.

3

What is the information block as a polynomial  $u(x)$?

	$u(x) = \alpha^3 \cdot x + x^2 + \alpha^4 \cdot x^3$,
	$u(x) = \alpha^3 + x + \alpha^4 \cdot x^2$,
	$u(x) = \alpha^4 + x + \alpha^3 \cdot x^2$.

4

What are the code symbols  $c_0, \hspace{0.05cm} \text{...} \hspace{0.1cm} ,\ c_6$  for the first coding step.

	$c_0 = \alpha^2$,
	$c_1 = \alpha^3$,
	$c_2 = \alpha^3$,
	$c_3 = 1$,
	$c_4 = \alpha^2$,
	$c_5 = \alpha^4$,
	$c_6 = 1$.

5

What is the binary code word?

	$\underline{c}_{\rm bin} = 100_|011_|011_|001_|110_|100_|001$,
	$\underline{c}_{\rm bin} = 011_|011_|001_|110_|100_|001_|100$,
	$\underline{c}_{\rm bin} = 100_|111_|0$.

6

Which statements apply to the second coding step?

	It holds  $u_0 = u_1 = u_2 = 0$.
	It holds  $u(x) = 1$.
	The code word  $\underline{c} ∈ \rm GF(2^3)$  consists of seven zero symbols.
	The binary code word   $\underline{c}_{\rm bin} ∈ \rm GF(2)$  consists of  $21$  zeros.

Solution

Solution

(1)  Correct is the  proposed solution 2:

• All information symbols here come from the Galois field  $\rm GF(2^3)$.  In binary notation,  each symbol is represented by three bits.

• Because of the Reed-Solomon code parameter  $k = 3$,  an information block consists of three information symbols:  $\underline{u} = (u_0, \, u_1, \, u_2)$.

• Accordingly,  the binary information block $\underline{u}_{\rm bin}$  contains exactly  $3 \cdot 3 = 9$  bits.

(2)  According to the table on the information page,  there is the following relationship between the coefficient vector and the exponential representation:

$$(110) \hspace{0.1cm} \Rightarrow \hspace{0.2cm} u_0 = \alpha^{4}\hspace{0.05cm},\hspace{0.6cm} (001) \hspace{0.1cm} \Rightarrow \hspace{0.2cm} u_1 = \alpha^{0} = 1\hspace{0.05cm},\hspace{0.6cm} (011) \hspace{0.1cm} \Rightarrow \hspace{0.2cm} u_2 = \alpha^{3}\hspace{0.05cm}. $$

• Correct are the  proposed solutions 1, 4, and 5.

• The information block in the first coding step is  $\underline{u} = (\alpha^4, \, 1, \, \alpha^3)$.

(3)  In polynomial representation,  the information block  $\underline{u} = (\alpha^4, \, 1, \, \alpha^3)$  results in:

$$u(x) = u_0 + u_1 \cdot x + u_2 \cdot x^2 = \alpha^{4} + x + \alpha^{3}\cdot x^2 \hspace{0.05cm}. $$

• Therefore the  proposed solution 3  is correct.

• The first proposed solution cannot be correct,  since the polynomial degree can be at most  $2$,  if all information and code symbols originate from  $\rm GF(2^3)$.

(4)  For the code symbols you get with the polynomial representation according to the specification page:

$$c_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{0} = 1) = \alpha^{4} + 1 + \alpha^{3}\cdot 1^2 = (110) + (001) + (011)= (100) = \alpha^{2} \hspace{0.05cm},$$

$$c_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{1}) = \alpha^{4} + \alpha + \alpha^{5}= (110) + (010) + (111) = (011) = \alpha^{3} \hspace{0.05cm},$$

$$c_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{2}) = \alpha^{4} + \alpha^{2} + \alpha^{7}= (110) + (100) + (001) = (011) = \alpha^{3} \hspace{0.05cm},$$

$$c_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{3}) = \alpha^{4} + \alpha^{3} + \alpha^{9}= (110) + (011) + (100) = (001) = 1 \hspace{0.05cm},$$

$$c_4 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{4}) = \alpha^{4} + \alpha^{4} + \alpha^{11} = \alpha^{4} \hspace{0.05cm},$$

$$c_5 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{5}) = \alpha^{4} + \alpha^{5} + \alpha^{13}= (110) + (111) + (101) = (100) = \alpha^{2} \hspace{0.05cm},$$

$$c_6 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u(x = \alpha^{6}) = \alpha^{4} + \alpha^{6} + \alpha^{15}= (\alpha^{2} + \alpha) + (\alpha^2 +1) + \alpha = 1 \hspace{0.05cm}.$$

• So,  correct are the  proposed solutions 1, 2, 3, 4, 7.

• The solutions to 5 and 6 are exactly reversed.

(5)  The  first proposed solution  is correct:

• This results from the result of the subtask  (4)  and the assignment

$$\alpha^2 ↔ 100, \ \alpha^3 ↔ 011, \ \alpha^3 ↔ 011, \ 1 ↔ 001, \ \alpha^4 ↔ 110, \ \alpha^2 ↔ 100, \ 1 ↔ 001. $$

• The last proposed solution cannot be correct,  since the binary code word must consist of  $n \cdot m = 7 \cdot 3 = 21$  bits.

• Even if you have determined only the result  $c_0 = \alpha^2$  in the subtask  (4),  you already know that the second answer cannot be correct.

(6)  The  statements 1, 3, 4  are correct:

• The bits  10, ... , 18  of the input sequence are all  $0$  and thus also the information symbols  $u_0, \ u_1, \ u_2 ∈ \rm GF(2^3)$.

• Correspondingly,  $u(x) = 0$,  so that all code symbols  $c_i = u(x = \alpha^i)$  correspond to the zero symbol  $($index: $0 ≤ i ≤ 6)$.

• The binary code sequence thus consists of  $n \cdot m = 21$  "zeros".