The supplement describes how to detect and correct single errors, and how to only detect but not correct two errors. This so-called double-error detection, single-error correction scheme is sometimes called DED-SEC. We are going to work through one.

Calculate the 8 bit string that includes: 4 data bits 0110 3 check bits p1, p2 and p4, and an overall parity bit calculated across the other 7

Answer should be writen here:

p1	p2	d1	p4	d2	d3	d4	pD

Background: The supplemental chapter on error correction covers this in the very last section before the exercises This mechanism works by adding one additional parity bit that checks EVERYTHING all the data and parity bits as a group. This additional DED bit is added to the end of the Hamming-encoded string. For a 12-bit number (which has 8 bits of data and 4 Hamming bits), you would add it to the end, a 13th bit. For a system that ships 4 bits of data at a time (like some memory systems, perhaps), youd need 3 Hamming bits to protect the 4 data bits, plus an 8th DED parity bit. That last scheme would look as follows:

1	2	3	4	5	6	7	8
p1	p2	d1	p4	d2	d3	d4	pD

When you received the transmitted package and needed to check the data integrity, youd revalidate the 3 parity bits by calculating C1, C2, C2, etc. to make sure theyre zero, and youd also XOR the entire string including the additional parity bit P to check for a double error. The table below shows how to interpret the results. C = 0 means that all of the check bits were zero. Otherwise C 0. Using check bits and an additional overall parity bit you can have four possible outcomes: If C = 0 & P = 0 No error occurred. All of the check bits are zero. If C 0 & P = 1 A single error occurred that can be corrected. If C 0 & P = 0 A double error occurred that is detected but cannot be corrected. If C = 0 & P = 1 An error occurred in the DED bit