The matrix G I4 A where G 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

The matrix G = [I_4 | A], where G = [1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 | 0 1 1 1 0 1 1 1 0 1 1 1] is a generator matrix in standard form for a [7, 4] that we denote by H_3, By Theorem binary code 1.2.1 a parity check matrix for H_3 is H = [\'A^T | I_3] = [0 1 1 1 1 0 1 1 1 1 0 1 | 1 0 0 0 1 0 0 0 1] This code is called the [7, 4] Hamming code. Find at least four information sets in the [7, 4] code H_3 from Example 1.2.3. Find at least one set of four coordinates that do not form an information set. Often in this text we will refer to a subcode of a code c. If c is not linear (or not known to be linear), a subcode of C is any subset of c. If c is linear, a subcode will be a subset of C which must also be linear; in this case a subcode of c is a subspace of c.

Solution

The minimum distance of Hamming’s (7,4) code is three.

Hence the 1-spheres centered on the codewords are disjoint.

Now the Hamming 1 sphere of any codeword must contain

        7 + 7

There are 16 codewords.

Hence the 16 Hamming 1-spheres with center at each of the 16

Code words must contain a total of 16X8=128.

0 1 1 1 1 0 0

H=10 1 1 0 1 0

1 10 1 0 0 1

                          Linear equations of Hamming’s (7,4) code:

                    C2+c3+c4+c5=0

                     C1+c3+c4+c6=0

                   C1+c2+c4+c7=0

No column of H can be all zeros,or else an error in the corresponding code vector position would not affect the syndrome and would be undetectable.

All columns of H must be unique.If two columns are Identical errors corresponding to these code word

Locations will be indistinguishable.

A matrix H is called a parity-check matrix for a linear code C if the columns of H form a basis for the dual code Ci.