Let $C\leq F^n$ be a linear code of length $n$, over the field $F$.
We say that the linear code $D$ is equivalent to $C$ if it differs by a fixed reordering of the vector coordinates. I.e., there exists $\sigma \in S_n$, such that $\sigma:C \to D$ is a bijection. Where the action is defined $\sigma (c_1,\dots ,c_n)=(c_{\sigma(1)},\dots c_{\sigma(n)})$.
Now, this is a natural definition of equivalence as it preserves many important properties of codes, for example, minimum distance.
My question is then, are there any interesting properties $X$ or theorems of the following form:
"Given a linear code $C$, there exists an equivalent code $D$ such that $X$ property holds"?
I.e., is there anything nice or useful that may not be true for our original code, but will be for one that is equivalent?
Thanks.
Some codes (typically, cyclic codes) have specific burst error detection properties.
For example, consider the cyclic Hamming $(7,4)$ code given by $g(x)=x^3+x+1$. Among its properties:
It has distance $d=3$, hence it can detect $t=2$ errors inside a block, and can correct $1$ error.
It's cyclic. Its codewords have a maximum span length of that of $g(x)$, i.e., $\ell=4$. Hence (in addition of the properties above) it can detect any burst error of length $\ell-1=3$ or less.
Properties $1$ are invariant under equivalence. Properties $2$ are not.
More in general, the probability of decoding error, assuming the model of independent errors, depends only on the weight distribution of the codewords (weight enumerator function) - and this is contant for equivalent codes. But if the errors are not independent the above is not true, and two equivalent codes can have different probability of decoding error.