Exercise 2.10Z: Code Rate and Minimum Distance

The inventors of the Reed-Solomon codes

The codes developed by "Irving Stoy Reed" and "Gustave Solomon" in the early 1960s are referred to in this tutorial as follows:

$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$

The code parameters have the following meanings:

$q = 2^m$ is an indication of the size of the Galois field ⇒ ${\rm GF}(q)$,
$n = q - 1$ is the code length (symbol number of a code word),
$k$ indicates the dimension (symbol number of an information block),
$d_{\rm min}$ denotes the minimum distance between two codewords. For any Reed-Solomon code, $d_{\rm min} = n - k + 1$.
No other code with the same $k$ and $n$ yields a larger value.

Hints:

The exercise belongs to the chapter "Definition and properties of Reed–Solomon Codes".
Information relevant to this exercise can be found on the "Code name and code rate" page.

Questions

$q \hspace{0.2cm} = \ $

$e \hspace{0.2cm} = \ $

$t \hspace{0.2cm} = \ $

$R \hspace{0.2cm} = \ $

$d_{\rm min} \ = \ $

$R \hspace{0.2cm} = \ $

$d_{\rm min} \ = \ $

$N_{3} \ = \ $

$N_{4} \ = \ $

Solution

(1) From the code length $n = 255$ follows $q \ \underline{= 256}$.

The code rate is given by $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$

The minimum distance is $d_{\rm min} = n - k +1 = 255 - 223 +1 \hspace{0.15cm}\underline {=33}\hspace{0.05cm}.$

This allows

$e = d_{\rm min} - 1 \ \underline{= 32}$ symbol errors can be detected, and.
$t = e/2$ (rounded down), so $\underline{t = 16}$ symbol errors can be corrected.

(2) The code $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$ is the binary representation of the ${\rm RSC} discussed in (1) \, (255, \, 223, \, 33)_{256}$ with exactly the same code rate $R \ \underline{= 0.8745}$ and also the same minimum distance $d_{\rm min} \ \underline{= 33}$ as this one. Here $8$ bits (1 byte) are used per code symbol.

(3) From $d_{\rm min} = 33$ follows again $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.

If exactly one bit is corrupted in each code symbol, this also means 16 symbol errors.
This is the maximum value that the Reed–Solomon decoder can still handle.

(4) The RS decoder can correct 16 corrupted code symbols,

whereby it does not matter whether in a code symbol only one bit or all $m = 8$ bits have been corrupted.
Therefore, with the most favorable error distribution, up to $N_4 = 8 \cdot 16 \ \underline{= 128}$ bits can be corrupted without the code word being incorrectly decoded.