Note that the rows of H will also be independent. In general, there are also sev

Question

Note that the rows of H will also be independent. In general, there are also several possible matrix in matrices for C. The next gives one of them when c a generator has standard form. In this theorem AT is the transpose of A Theorem .2.1 vo ua l Alis a generator matrix for the In.klcode c in landore2 then [--A El a pariry check matrix for C, is Proof: we clearly have HGT--AT AT-o. use is contained in the kernel of the linear transformation x HxT. As H has rank n k, th is linear transformation has kemel of k, which is also the dimension of c. The result follows. dimension Exercise i Prior to the statement of Theorem 1.2.1. it was noted that the rows of the The x n parity check matrix H (.1) are independe Why is that so? Hint: map x Hx is a linear transformation nt. algebra, what is the rank from Fn F-1 with kemel c. From linear of H? Example 1.2.2 he simplest way to encode information in order to recoverit in the presence of noise is to repeat each message symbol a fixed information is binary with symbols number of times. Suppose that our for instance n 7, then from the field F2.and we repeat each symbol n times. If whenever we want to send a we send 0000000, and whenever we want to send a 1 we send 1111111. If at most three e are made in transmission and if we decode by "majority vote," then we can correctly determine the information symbol, or 1. n general, our code c is the In, 11binary linear code consisting of the two can 0 and 1 11... 1 and is called binary repetition code of n. This code correct the length up to e l(n 1)/2j emors: if at most e errors are made in a received vector. then the majority of coordinates will be correct, and hence the original sent codeword can be recovered. If more than e errors are made, these errors cannot be corrected. However, this code can detect n -1 errors, as received vectors with between 1 and n 1 errors will 1.3 Dual codes definitely not be codewords. A generator matrix for the repetition code is which is of course in standard form. The corresponding parity check matrix from Theorem 1.2.1 is The first coordinate is an information set and the last n 1 coordinates form a redundancy set X3 Exercise 2 How many information sets are there for the In, 11 repetition code of Example 1.2.2?

Explanation / Answer

Ignoring the identity parts Ik and In-k, the augmented matrices G and H have the following information parts

G = [A] and H = - AT, for the code in standard form.

Now, we have

HGT = - AT + (A)T = 0 [With respect to operation of addition of matrices.]

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