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**Hamming Code Details**

Our Hamming Code demonstration uses a (7, 4, 3) Hamming Code as described by Shannon in detail. Let c, e, f, and g be a block of 4 symbols from the source, represented as integers in the field with 2 elements.

We assign:

a = c + e + g

b = c + f + g

d = e + f + g

and transmit the codeword abcdefg. The noise agent then reverses one bit, or no bits at all. Upon receipt, the receiver computes the following:

A = d + e + f + g

B = b + c + f + g

C = a + c + e + g

Without noise, A, B, and C should all be zero. If they are not all zero, the 7 different possible combinations
reveal which bit was reversed. In fact, the binary number ABC gives the position of the incorrect bit. This
can be computed directly, or see *Elements of Information Theory* by Cover and Thomas for a derivation from
Linear Algebra.

Throughout the above discussion, we are always dealing with the one possible two-element field. Note, however, that in the demonstration, we used different alphabets to represent its elements: {2, 4} for the source and the destination, and {0, 1} for the channel itself.

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