Everything over F_2. Let us  define Hamming norm of polynom |p(x)| = number of non-zero monoms.

Respectivly for a pair of polynoms |[p ; g]| = |p| +|g|.

Consider linear map $F_2[x] \to F_2[x] \oplus F_2[x] $ given by
$p(x) \mapsto [ p(x)(x^2+1) ; p(x) (x^2+x+1)] $.

**Question**
Is it true that minimal Hamming norm  in the image  of the map above equal to 5 ?
Of course, delete [0; 0] from the image.

It is clearly not more than 5, since take p(x)=1,
then $1 \mapsto [ x^2+1; x^2+x+1] $. and $|x^2+1| =2$  $x^2+x+1 = 3$ , So 2+3 =5.

By brute force search for p(x): deg p <16 the answer is 5. 

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This is toy model for convolutional error correcting codes.

See the question http://mathoverflow.net/questions/102434/given-g1x-g2x-minimize-over-px-hamming-weight-of-pxg1-pxg2x-o

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