Fix polynoms g1(x), g2(x) over F_2[x].

**Question:** How to find minimum over polynoms p(x) of the:

*HammingWeight(p(x) g1(x) ) +  HammingWeight(p(x) g2(x) )* ? 

By HammingWeight of polynom I mean number of non-zero monoms, let me denote by || *||

Trivial estimate: minimum <= || g1(x)|| +  || g2(x)||. Proof just put p(x) =1.

Numerical observation: apparently there are polynoms g1,g2 where this estimate is exact.
Can this be true ?

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**Modified question 1** If I put restriction deg(p(x)) < N with N>> deg(gi) will it change minimum ? if yes what can be said about it ? 

**Modified question 2** If I put restriction deg(p(x)) < N and moreover
will consider multiplication in the factor F_2[x]/ (x^N-1) will it change minimum ? if yes what can be said about it ? 

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**Error-correcting codes formulation** 
The question is how to calculate minimal distance of [non-recursive convolutional code][1]  ?

If I put restriction deg(p) < N this corresponds to various truncations of them.
In particular working with F_2[x]/ (x^N-1) corresponds to "tail-biting".

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PS

This is completely rewritten version of the previous question.


  [1]: http://en.wikipedia.org/wiki/Convolutional_code#Recursive_and_non-recursive_codes