fuckchadsandstaceys
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The world is so fucking unfairI felt my heart sink into my stomach watching this
[UWSL]H. Coding Theory: Generator Matrix and Parity-Check Matrix of a Code[/UWSL]
[UWSL]For the reader who does not know the subject, linear algebra will be developed in Chapter 28. However,[/UWSL]
[UWSL]some rudiments of vector and matrix multiplication will be needed in this exercise; they are given here:[/UWSL]
[UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]vector[/UWSL][UWSL] [/UWSL][UWSL]with[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]components is a sequence of[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]numbers: ([/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL]). The[/UWSL][UWSL] [/UWSL][UWSL]dot product[/UWSL][UWSL] [/UWSL][UWSL]of two[/UWSL]
[UWSL]vectors with[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]components, say the vectors[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL]) and[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]b[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL]n[/UWSL][UWSL]), is defined by[/UWSL]
[UWSL]a[/UWSL][UWSL] [/UWSL][UWSL]·[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL]) · ([/UWSL][UWSL]b[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL]n[/UWSL][UWSL]) =[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL]b[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL]b[/UWSL][UWSL]2[/UWSL][UWSL] [/UWSL][UWSL]+ · · · +[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL]b[/UWSL][UWSL]n[/UWSL]
[UWSL]that is, you multiply corresponding components and add. For example,[/UWSL]
[UWSL](1,4, –2, 3) · (6, 2, 4, –2) = 1(6) + 4(2) + (–2)4 + 3(–2) = 0[/UWSL]
[UWSL]When[/UWSL][UWSL] [/UWSL][UWSL]a · b[/UWSL][UWSL] [/UWSL][UWSL]= 0, as in the last example, we say that[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL] [/UWSL][UWSL]and[/UWSL][UWSL] [/UWSL][UWSL]b[/UWSL][UWSL] [/UWSL][UWSL]are[/UWSL][UWSL] [/UWSL][UWSL]orthogonal[/UWSL][UWSL].[/UWSL]
[UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]matrix[/UWSL][UWSL] [/UWSL][UWSL]is a rectangular array of numbers. An “[/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]by[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]matrix” ([/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]×[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]matrix) has[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]rows and[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL]
[UWSL]columns. For example,[/UWSL]
[UWSL]is a 3 × 4 matrix: It has three rows and four columns. Notice that each row of[/UWSL][UWSL] [/UWSL][UWSL]B[/UWSL][UWSL] [/UWSL][UWSL]is a vector with four[/UWSL]
[UWSL]components, and each column of[/UWSL][UWSL] [/UWSL][UWSL]B[/UWSL][UWSL] [/UWSL][UWSL]is a vector with three components.[/UWSL]
[UWSL]If[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]is any[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]×[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]matrix, let[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]be the[/UWSL][UWSL] [/UWSL][UWSL]columns[/UWSL][UWSL] [/UWSL][UWSL]of[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL]. (Each column of[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]is a vector with[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL]
[UWSL]components.) If[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]is any vector with[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]components,[/UWSL][UWSL] [/UWSL][UWSL]xA[/UWSL][UWSL] [/UWSL][UWSL]denotes the vector[/UWSL]
[UWSL]xA[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]x[/UWSL][UWSL]·[/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL]·[/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL]·[/UWSL][UWSL]a[/UWSL][UWSL]n[/UWSL][UWSL])[/UWSL]
[UWSL]That is, the components of[/UWSL][UWSL] [/UWSL][UWSL]xA[/UWSL][UWSL] [/UWSL][UWSL]are obtained by dot multiplying[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]by the successive columns of[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL]. For[/UWSL]
[UWSL]example, if[/UWSL][UWSL] [/UWSL][UWSL]B[/UWSL][UWSL] [/UWSL][UWSL]is the matrix of the previous paragraph and[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]= (3,1, –2), then the components of[/UWSL][UWSL] [/UWSL][UWSL]xB[/UWSL][UWSL] [/UWSL][UWSL]are[/UWSL]
[UWSL]that is,[/UWSL][UWSL] [/UWSL][UWSL]xB[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]−[/UWSL][UWSL]7, 3,[/UWSL][UWSL] [/UWSL][UWSL]−[/UWSL][UWSL]15, 8).[/UWSL]
[UWSL]If[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]is an[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]×[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]matrix, let[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL](1)[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL](2)[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]([/UWSL][UWSL]m[/UWSL][UWSL])[/UWSL][UWSL] [/UWSL][UWSL]be the[/UWSL][UWSL] [/UWSL][UWSL]rows[/UWSL][UWSL] [/UWSL][UWSL]of[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL]. If[/UWSL][UWSL] [/UWSL][UWSL]y[/UWSL][UWSL] [/UWSL][UWSL]is any vector with[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]components,[/UWSL]
[UWSL]Ay[/UWSL][UWSL] [/UWSL][UWSL]denotes the vector[/UWSL]
[UWSL]Ay[/UWSL][UWSL] [/UWSL][UWSL]= ([/UWSL][UWSL]y[/UWSL][UWSL] [/UWSL][UWSL]·[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL](1)[/UWSL][UWSL],[/UWSL][UWSL] [/UWSL][UWSL]y[/UWSL][UWSL] [/UWSL][UWSL]·[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL](2)[/UWSL][UWSL], ...,[/UWSL][UWSL] [/UWSL][UWSL]y · a[/UWSL][UWSL]([/UWSL][UWSL]m[/UWSL][UWSL])[/UWSL][UWSL])[/UWSL]
[UWSL]That is, the components of[/UWSL][UWSL] [/UWSL][UWSL]Ay[/UWSL][UWSL] [/UWSL][UWSL]are obtained by dot multiplying[/UWSL][UWSL] [/UWSL][UWSL]y[/UWSL][UWSL] [/UWSL][UWSL]with the successive[/UWSL][UWSL] [/UWSL][UWSL]rows[/UWSL][UWSL] [/UWSL][UWSL]of[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL]. (Clearly,[/UWSL]
[UWSL]Ay[/UWSL][UWSL] [/UWSL][UWSL]is not the same as[/UWSL][UWSL] [/UWSL][UWSL]yA[/UWSL][UWSL].) From linear algebra,[/UWSL][UWSL] [/UWSL][UWSL]A[/UWSL][UWSL]([/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]y[/UWSL][UWSL]) =[/UWSL][UWSL] [/UWSL][UWSL]Ax[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]Ay[/UWSL][UWSL] [/UWSL][UWSL]and ([/UWSL][UWSL]A[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]B[/UWSL][UWSL])[/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]=[/UWSL][UWSL] [/UWSL][UWSL]Ax[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]Bx[/UWSL][UWSL].[/UWSL]
[UWSL]We shall now continue the discussion of codes begun in Exercises F and G of Chapter 3. Recall that[/UWSL]
[UWSL]is the set of all vectors of length[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]whose entries are 0s and 1s. In Exercise F, page 32, it was shown[/UWSL]
[UWSL]that[/UWSL][UWSL] [/UWSL][UWSL]is a group. A[/UWSL][UWSL] [/UWSL][UWSL]code[/UWSL][UWSL] [/UWSL][UWSL]is defined to be any subset[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL] [/UWSL][UWSL]of[/UWSL][UWSL] [/UWSL][UWSL]. A code is called a[/UWSL][UWSL] [/UWSL][UWSL]group code[/UWSL][UWSL] [/UWSL][UWSL]if[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL] [/UWSL][UWSL]is a[/UWSL]
[UWSL]subgroup[/UWSL][UWSL] [/UWSL][UWSL]of[/UWSL][UWSL] [/UWSL][UWSL]. The codes described in Chapter 3, as well as all those to be mentioned in this exercise,[/UWSL]
[UWSL]are group codes.[/UWSL]
[UWSL]An[/UWSL][UWSL] [/UWSL][UWSL]m[/UWSL][UWSL] [/UWSL][UWSL]×[/UWSL][UWSL] [/UWSL][UWSL]n[/UWSL][UWSL] [/UWSL][UWSL]matrix[/UWSL][UWSL] [/UWSL][UWSL]G[/UWSL][UWSL] [/UWSL][UWSL]is a[/UWSL][UWSL] [/UWSL][UWSL]generator matrix[/UWSL][UWSL] [/UWSL][UWSL]for the code[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL] [/UWSL][UWSL]if[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL] [/UWSL][UWSL]is the group generated by the rows of[/UWSL]
[UWSL]G[/UWSL][UWSL]. For example, if[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]is the code given on page 34, its generator matrix is[/UWSL]
[UWSL]You may check that all eight codewords of[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]are sums of the rows of[/UWSL][UWSL] [/UWSL][UWSL]G[/UWSL][UWSL]1[/UWSL][UWSL].[/UWSL]
[UWSL]Recall that every codeword consists of information digits and parity-check digits. In the code[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]the[/UWSL]
[UWSL]first three digits of every codeword are information digits, and make up the[/UWSL][UWSL] [/UWSL][UWSL]message;[/UWSL][UWSL] [/UWSL][UWSL]the last two digits[/UWSL]
[UWSL]are parity-check digits.[/UWSL][UWSL] [/UWSL][UWSL]Encoding[/UWSL][UWSL] [/UWSL][UWSL]a message is the process of adding the parity-check digits to the end of[/UWSL]
[UWSL]the message. If[/UWSL][UWSL] [/UWSL][UWSL]x[/UWSL][UWSL] [/UWSL][UWSL]is a message, then[/UWSL][UWSL] [/UWSL][UWSL]E[/UWSL][UWSL]([/UWSL][UWSL]x[/UWSL][UWSL]) denotes the encoded word. For example, recall that in[/UWSL][UWSL] [/UWSL][UWSL]C[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]the[/UWSL]
[UWSL]parity-check equations are[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]4[/UWSL][UWSL] [/UWSL][UWSL]=[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]3[/UWSL][UWSL] [/UWSL][UWSL]and[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]5[/UWSL][UWSL] [/UWSL][UWSL]=[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL] [/UWSL][UWSL]+[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]3[/UWSL][UWSL]. Thus, a three-digit message[/UWSL][UWSL] [/UWSL][UWSL]a[/UWSL][UWSL]1[/UWSL][UWSL]a[/UWSL][UWSL]2[/UWSL][UWSL]a[/UWSL][UWSL]3[/UWSL][UWSL] [/UWSL][UWSL]is[/UWSL]
[UWSL]encoded as follows:[/UWSL]
I felt my heart sink into my stomach watching this
At least i can cope by saying she looks used up (like all white foids do)The world is so fucking unfair
the only looks I get from girls is looks of disgust
Not that much taller than the gfHow tall is he? He looks subhuman otherwise, i don't know how he managed to get a white foid. Ricefuel for lifecels.
JBR
Just be rice
Right? His face is maybe 5/10 at best. But he is a tallfag and it hard carries him. Height = Life. Fuck I hate being a short nigger.Why does this yellow subhuman have a girlfriend and I don't? Teach me your yellow tricks, shifu.
I felt my heart sink into my stomach watching this
Spoken like the Chadpreet you areJBR
Just be rice
Same,toilets really are demonic elements.the only looks I get from girls is looks of disgust