I am afraid answer is trivial for large N and equal just to 2.
The same argument is in http://mathoverflow.net/questions/101374/
- one can find binom x^l-1 divisible by all polynoms v_i.
That would mean answer is TWO.

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**Mathematical formulation** Fix some vectors $v_1, v_2 \in F_2^k$ ,
take some number N>k and consider circulant NxN matrices  $C(v_1), C(v_2) $ which are defined
by these vectors (put zeros on diagonals which are further then k-th diagonal). 

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by 
$x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight,
denote this weight by $dMin(N)$.

**Question** How does $dMin(N)$ behave depending on N?

**Comments** Of course, everything depends on $v_i$, this makes the question vague,
so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

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I am not sure my description of C(v) is clear, let me give another.
Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. 
Consider the operator of multiplication on this polynom acting in a factor
space $F_2[x]/(x^N-1)$ - this is exactly C(v).

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Numerical experiment.

I took three vectors corresponding to binary form of octal numbers 

    G4 = 133;       % 1+D^2+D^3+D^5+D^6

    G7 = 171;       % 1+D+D^2+D^3+D^6

    G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:24, here is the answer:

(E.g.  9     9    10    10    12    12    13    13    12    13    15  15    15    15    15)

I am using MatLab build-in functions so for conv.codes there is certain chance
my math. interpretation does not fully correspond to them, but hope it is Okay.

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**Error-correcting codes formulation** 
The question is how the minimal distance of the "tail-bited" [non-recursive convolutional code][1] behave depending on block lenght N ?

**Reminder** Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

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Pay attention this is NOT kind of code considered in here:

http://mathoverflow.net/questions/101374/how-many-k-nomials-of-deg-n-divisible-by-x16x12x5-1-spectrum-of-crc-16/101468#101468

Some related question:

http://mathoverflow.net/questions/101471/what-is-matrix-a-such-that-hamming-weight-of-x-ax-is-maximal-min-distance

But not I fix the class of codes to be convolutional.



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