Consider polynoms f(x) g(x) of degree at most n. (I am mostly interested about F_2[x]).

Let us multiply them by arbitrary polynoms p(x) i.e. consider ideal (p f , p g) in $F_2[x]\oplus F_2[x]$.

Let us delete zero from this ideal and calculate N(f,g) - Hamming distance of I$\backslash${0} to {0}. (Here Hamming distance is just the number of non-zero monoms).

**Questions** What is Max_{f,g of degree n} N(f,g) ? (Or at least some bounds on it ?)
What polynoms give this max ?

What seems not trivial:

It is clear that N(f,g) <= |f| + |g| (take p(x) =1 ) .

So it seems that we should take |f| and |g| of maximal possible Hamming norm i.e. g = f = x^n+x^{n-1} + ... x^2 +x +1

But it clear that N(f,g) =4 for this choice of f,g - i.e. very small. (Proof - just multiply them by p(x) = (x+1)).

So these two effects are fighting each other - we want to take |f| , |g| big, but multiplication by x+1 will spoil N(f,g) if these norms are too big....

Related question

From coding theory viewpoint

map: p(x) -> (p f , p g) is encoder (non-recursive convolutional code of rate 1/2).

I am asking what is "best possible" code for deg f,g

Examples

if n = 2, then f=x^2 +1 , g= x^2+x+1 gives max which equal to 5

(see Multiplication by polynomials x^2+1 ; x^2+x+1. Does minimal Hamming norm of image equal to 5 ? )

If n=3 then max N(f,g) = 6 and can be realized by many choices e.g. f=x^3+x^2+1, g=x^3+x+1 or f=x^3+x^2+x+1, g=x^3+x+1 (if I am not mistaking).

If n=4, then possibly(?) max N(f,g) = 8 for e.g. f=(x+1)(x^3+1) ; g=(x+1)(x^3+x+1) ------ NOT TRUE : take p(x)=x^2+x+1, we get N(f,g)<=6 for these pols. It might that max N(f,g) = 7. achieved on e.g. f=x^4+x^3+x^2+1 g=x^4+x+1