I need help understanding the definitions and context for a homework question:
Consider a 3 by 7 matrix A over GF(2) containing distinct columns. The row space C of A is the
subspace over GF(2) generated by the 3 rows. (Extra note: This is a “simplex” code [7,3] with generator
matrix A. It is closely related to a certain “Hamming” code [7,4].)
Would the above mean that, for instance I have a matrix that has unique columns and elements in GF(2):
$A=
\begin{matrix}
0 & 0 & 0& 0& 1 & 1 &1\\
0 & 0 & 1& 1& 0 & 0 &1\\
0 & 1 & 0& 1& 0 & 1 &0
\end{matrix}
$
the row space would then be:
$C=[0000111, 0011001, 0101010]$
The next parts of the questions needs me to know about the
weight distribution of C, weight of a vector, distance between words
Could anyone explain what those words mean in this context?
Best Answer
When it says "having distinct columns", it most likely means "distinct nonzero columns" (this is the matrix commonly used in relation to the Hamming code). There are exactly 7 possible nonzero columns in $\mathbb{F}_{2}^{3}$, so the matrix should be $$\begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{bmatrix}$$ (in some order; your matrix is fine if you get rid of the all 0 column and replace it with the all 1 column).
Then the row space is the set of all possible linear combinations of these three rows; the row space is $3$-dimensional, so contains a total of $8$ vectors.
The weight of a codeword is the number of nonzero entries, so $01001011$ would have weight 4.
The distance between two codewords is the weight of their difference, or equivalently, the number of places where they are not equal; so $00001111$ and $11001100$ would be at distance 4.
The weight distribution is the number of codewords of each weight. It is commonly represented as a polynomial, where for example the term $3x^4$ would mean there are 3 codewords of weight $4$.