Question

Question 16: Show that a BCH code with t = 1 is actually a Hamming code (see page 490).Question 16: Show that a BCH code with is actually a Hamming code (see page 490).

Short answer

Answer

It has been proved that a BCH code with t = 1 is actually a Hamming code.

Step-by-step solution

Consider the results used

If C is a linear code with parity-check matrix H. Then C corrects single errors if and only if the rows of H are distinct and nonzero.

Write the given data

Given BCH code with t =1.

Prove that it is a Hamming Code

A Hamming Code is one whose parity check matrix H is matrix whose rows are the 7 distinct nonzero elements of B(3).

The BCH code constructed with t = 1 and r = 3 has $n=2^r-1=7$and field K of $2^r=8$elements. For example $k=\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^2+x+1\right)$}\right.$has generator with minimal polynomial $m_1\left(x\right)=x^3+x+1$.

As before the minimal polynomial for is also , so that . Then $m=\mathrm{deg} g\left(x\right)=3$and$\$k=n-m=4\$$.

Therefore we have a BCHcode. By the theory of BCHcodes this one corrects

Single errors.

Therefore, by the above mentioned result the parity check matrix Hmust have rows which are distinct and nonzero.

However, this His a 7 × 3 matrix so that all 7 of the nonzero elements of B(3) must occur as rows of H, and thus a Hamming Code is obtained.