Dan Ernst Improving Reliability We used parity to determine when a memory bit failed. We can protect buses from transmission failures using parity/ECC. Detected with parity: 1 bit errors Corrected with ECC (Error correcting codes)

Dan Ernst Parity If any two different valid datum, in memory or on a bus, differ by at least 2 bits: It is easy to detect if one bit fails since a one bit failure will result in an invalid data value. How can we make sure any good data differs by at least two bits?

Dan Ernst Why parity works to detect one bit errors… All valid (and unique) data encoding must differ by at least one bit. –Argument: if they don’t they aren’t unique. Pick any two values: –If they differ by more than one bit, a single bit error will not turn one into the other –If they differ by one bit then if we count the number of 0 bits in their encoding, one of the encodings will have an odd number of 0s, the other will have an even number and therefore their parity bits MUST also differ, so they will also differ by two bits. So we can’t change one valid encoding into another by changing only 1 bit!

Dan Ernst Example Two number: and Add odd parity bits: and Now these numbers differ by 2 bits What if they already differ by more than one bit? –No problem, a 1-bit error can’t turn one into the other

Dan Ernst Error Correcting Codes If any two different valid datum, in memory or on a bus, differ by at least 3 bits: It is easy to detect and correct if one bit fails since a one bit failure will result in an invalid data value and we know which valid data value is only one bit away. How can we make sure any good data differs by at least three bits?

Dan Ernst Error Correcting Codes Use multiple parity bits, each computing parity over a different set of data bits. Each data bit is used to calculate parity by a different combination (or permutation) of 2 or more parity bits. –data bit 0 may be used in the calculation of parity bits 1 and 2, –while data bit 1 is used by parity bits 1 and 3. When a parity bit is flipped, only its parity calculation will be wrong.

Dan Ernst ECC on 4 bits of data Data bit 0 is used by parity 0 and 1 Data bit 1 is used by parity 0 and 1 and 2 Data bit 2 is used by parity 0 and 2 Data bit 3 is used by parity 1 and 2 –P0 = odd_parity (D0, D1, D2) –P1 = odd_parity (D0, D1, D3) –P2 = odd_parity (D1, D2, D3)

Dan Ernst Calculating ECC (4 bits) D 3 D 2 D 1 D 0 P 2 P 1 P 0

Dan Ernst Calculating ECC (4 bits) D 3 D 2 D 1 D 0 P 2 P 1 P

Dan Ernst Calculating ECC (4 bits) D 3 D 2 D 1 D 0 P 2 P 1 P

Dan Ernst Calculating ECC (4 bits) D 3 D 2 D 1 D 0 P 2 P 1 P

Dan Ernst Test question The following 4 bit data value is encoded with ECC (as shown in class). Unfortunately it has a 1 bit error, fix it! D 3 is used by P 2 and P 1 P 2 P 1 P 0 D 3 D 2 D 1 D 0

Dan Ernst Test question Solution The flipped bit is D 3, which should be a D 3 is used by P 2 and P 1 P 2 P 1 P 0 D 3 D 2 D 1 D 0

Dan Ernst How many ECC parity bits (P) do you need for N bits of data? You use 1 ECC parity bit pattern for each data bit error. –N bit patterns for fixing 1 bit errors Plus 1 more pattern for each parity bit –P bit patterns Plus 1 pattern for “correct value” N + P + 1 = 2 P

Dan Ernst How many ECC bit do you need for a 78 bit bus? 78 + P + 1  2 P Patterns I need to decide which bit is wrong if 1 bit is flipped  Unique pattern I can represent with P ECC parity bits 2 6 = 64 (too small) 2 7 = 128 (which is > ) So, 7 ECC parity bits needed

Dan Ernst Question Prove that ECC has at least 3 bits different in any two representations. Answer: –if they are different, then they must differ by at least 1 data bit. –Each data bit is covered by at least 2 parity bits Those parity bits must now differ since one must have an odd number of data 1’s and the other even.

Dan Ernst Question How do you decide which bit is wrong? If only one parity bit is the wrong parity, then that parity bit has been corrupted –Because and data bit is checked by at least 2 parity bits a data bit failure will cause 2 or more parity bit errors. If two or more parity bits are wrong, then the pattern of the parity bits that fail uniquely identify the corrupted data bit.