Weight of polynomial over finite field

Question

For $\alpha \in F_{p}$, $\alpha \ne 0$, the polynomial $f(x) = (x+\alpha)^k$ has weight $k+1$, when $k < p$ (the weight is number of non-zero coefficients). This is since both $\binom{k}{i}$ and $\alpha^i$ are $\not\equiv 0 \pmod{p}$ for $0 \le i \le p-1$ . Thus all $k+1$ coefficients of $f$ are non-zero (for the next possible exponent, $p$, we have $(x+\alpha)^p = x^p + \alpha^p$ which has weight $2$).

What also seems to be true is that if we look at $(x+\alpha)^k \cdot g(x)$,and keep the degree of $g$ small (i.e. $\le p-k-1$ so that the product has degree $<p$) then the weight of $(x+\alpha)^k \cdot g(x)$ is at least k+1.

Can anyone point me to a proof of this (if it's true)?

Answer 1

What about $p=5$ with $k=1$ or $k=2$? Try $f(x)=x+1$ and $g(x)=x-1$, or $f(x)=(x+1)^2$, and $g(x)=(x-1)^2$.

Answer 2

Because $f(x)$ and $f(-\alpha x)$ have the same weight we can, without loss of generality, assume that $\alpha=-1$. In that case you are looking at "classical" repeated-root cyclic codes of length $p$ over $\Bbb{F}_p$ generated by $(x-1)^k$. Your claim is equivalent to stating that such codes are MDS. This is known to be the case. You will probably benefit from a study of

• Karl-Heinz Zimmermann, "On generalizations of repeated-root cyclic codes" from March 1996 issue of IEEE Transactions on Information Theory.

He attributes this result independently to Berman ('67) and Assmus & Mattson ('69).