Equivalent codes

Definition 3.4.17   If $G$ is any matrix with entries in a field $F$ then replacing any row of $G$ by (a) its sum with another row, or (b) its scalar multiple with any non-zero element of $F$, is called an elementary row operation. If $G$ is any matrix with entries in a field $F$ then (a) swapping any two columns, or (b) replacing any column of $G$ by its scalar multiple with any non-zero element of $F$, is called a simple column operation.

Lemma 3.4.18   If $G$ is a generating matrix for a linear code $C$ then so is $G'$, where $G'$ is any matrix obtained from $G$ by an elementary row operation.

proof: The rows of $G'$ still form a basis for the vector space $C$ over $F$. $\Box$

Definition 3.4.19   If $G$ is a generating matrix for a linear code $C$ and $C'$ is the linear code with generating matrix $G'$, where $G'$ is any matrix obtained from $G$ by elementary row operations or simple column operations, then we say that the codes $C$ and $C'$ are equivalent.

Example 3.4.20   For the ISBN code,

$$G=\left( \begin{array}{cccccccccc} 1&0&0&0&0&0&0&0&0&1\\ ... ...0&1&0&1&0&0&0&0\\ 1&1&1&0&1&0&0&0&0&0 \end{array} \right)$$

is a generating matrix. By the above lemma, so is

$$G'=\left( \begin{array}{cccccccccc} 1&0&0&0&0&0&0&0&0&1\\... ...&1&0&1&0&0&0&0\\ 1&1&1&0&1&0&0&0&0&0 \end{array} \right),$$

where we replaced the second row by its sum with the first row. However, if we swap the first two columns of $G$, to get

$$G''=\left( \begin{array}{cccccccccc} 0&1&0&0&0&0&0&0&0&1\... ...0&1&0&1&0&0&0&0\\ 1&1&1&0&1&0&0&0&0&0 \end{array} \right)$$

then $G''$ generates a code $C''$ which is different from $C$. The codes $C$ and $C''$ are equivalent.