Hamming Error Correcting Code

Michael Wimble

6026 Underwood Av SW

Cedar Rapids IA 52404

One of the most frustrating aspects ofcomputers is that they make errors. Largecomputers have ample redundancy and errorcorrecting hardware to make these errorsvirtually nonexistent, but owners of smallercomputers must typically live with theproblem. These errors are not all due tounreliable hardware; they are also caused bynoisy environments, line crosstalk, powerfluctuations, thermal variations and so on.To meet these problems several tech niqueshave been developed. One of them is calledHamming codes.

The use of Hamming codes is analogousto the use of the common parity bit. Asingle parity bit merely provides detectionof single bit errors, however. In contrast,the Hamming code described herein cor-rects single bit errors and detects doublebit errors. An ideal use for Hamming codes isin cassette recording where single bit drop-outs due to tape inconsistencies are common.Larger computers also use Hamming codesto detect and correct memory errors.

Alas, as with all real systems, you don'tget something for nothing. To record eightbits of data without any error detection orcorrection scheme requires only eight bitsof memory space. To use the commonparity bit approach adds only one more bitof data: nine bits of memory space. Butthe Hamming code described here requireseight extra bits of memory space for everyeight bits of data recorded. There are otherHamming codes available which use con-siderably less extra memory space per dataspace, but this one is particularly appro-priate for m icrocom pu ters and for cassetterecording, the most frequent source of errorsin a typical hobbyist microcomputer system.

No attempt will be made to discuss themathematics behind Hamming codes. Thereader is referred to any good book on datatransmission or error correcting codes formore information.

Building the Hamming Code

Building the Hamming code informationis as simple as a table lookup. Every byte(eight bits) of data is recorded a nybble(four bits) at a time. Also, each group offour bits of data has four bits of Hammingdata appended so that each byte of data tobe recorded actually requires two bytes ofstorage.

Recording a byte of data is straight-forward. Take the leftmost (most significant)nybble of data and record the correspondingbyte of data shown in table 1. Next take therightmost or least significant nybble of dataand, again, record instead the correspondingbyte from table 1.

For example, to record the data byte,hexadecimal 1 F, perform the following:

• Extract the leftmost nybble (hexa-decimaI01).

• Replace with corresponding byte fromtable 1 (hexadecimal E1).

• Record the byte.• Extract rightmost nybble (hexadecimal

OF)• Replace with corresponding byte from

table 1 (hexadecimal FF).• Record the byte.

Thus the single hexadecimal byte 1 F isactually recorded as the two hexadecimalbytes E1 FF. If it is not already obvious,each byte in table 1 has the actual data inits rightmost nybble and error correctingdata in its leftmost nybble.

Reconstructing the Data

We are now ready to see how the Ham-ming code functions. As each byte of storeddata is retrieved, with or without errors,four parity bits are constructed which giveinformation as to the correctness of theretrieved byte. Using these parity bits, anyrequired corrections are made to the re-

The timer processor scans the file ofdelay timer records (table 7) to processthose records which are active. The func-tion of the delay timer record is to providea delay of a preset number of seconds beforeprocessing an event record. To demonstratehow this function works, let's assume thatone wishes to activate an audible alarm 50seconds after a particular sensor has beentripped. You would structure the event andresponse records so that the tripping of thesensor would activate the delay timer recordyou were associating with this event. Theactivation of the delay timer causes theactive flag to be reset and the timer activa-tion value (bytes 2 and 3) to be transferredto the delay time. In our example this willcause the value 50 to be loaded into bytes 0and 1 of our delay timer value.

As each real time clock interrupt causesthe timer processor to scan the file of delaytime records, it will find that our record isactive. Once the processor determines that arecord is active it will decrement the currentdelay time remaining and then check to seeif the time remaining is zero. If there is stilldelay time remaining (time greater thanzero) no further action is taken.

However, if there is no time remaining(time equals zero) the active flag is set, thecurrent registers saved, the address of theevent record is extracted from the delaytime record, and control transferred to theevent processor. The sequence of operationsperformed from this point will directlyresult in the audible alarm being turned on.

The processing of the records in the timeof day file (see table 6 for record format) isperformed upon the completion of all delaytime records. Time of day records are proc-essed in a manner very similar to thatdescribed for delay timer records. When anactive record is encountered during the scanof the time of day file, the current time ofday being maintained by the system will becompared to the time of day specified inbytes 0 and 1 of the record. Should thesetimes be identical, the records flag is set to 1and the processing of the event record speci-fied in bytes 2 and 3 is initiated as describedabove. After servicing all records in the timeof day file, control is returned to the inter-rupted modules.-

Next Month: Part 3 of the Compu-terized Security System illustrates thedesign of some of the sensors, giveslistings of a few of the control modules,and describes the design of the remotesensor display panel.

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Data Nybbleo1234567

Hamming Byte00E172938455C627D839AA486C8D1EFF

Table 7: Table for transforming data nybblesinto Hamming code bytes for storage.

trieved byte after which the original nybbleof data is extracted.

Parity bit 4 (P4), the most significantparity bit, is simply the parity of the entire

bit retrieved byte. If the 8 bit data wordnas an odd number of 1s, then P4 is setto 1. Similarly, if the 8 bit data word has aneven number of 1s, P4 is set to O. To formparity bit 3 (P3), the 8 bit data word is firstANDed with hexadecimal 27 and the 8 bitparity of the result determines P3 just as P4was calculated. Likewise parity bit 2 (P2) isformed by ANDing the retrieved dataword with hexadecimal 4B; parity bit 1 (Pl),the least sign ificant parity bit, is formedfrom ANDing the retrieved data word withhexadecimal 80. Table 2 shows a samplecalculation of the parity bits when hexa-decimal is retrieved.

Error detection and correction is per-formed by detecting one of three cases:

• If all four parity bits are 0, the re-trieved data word is correct. Theoriginal nybble of data is correctlyformed in the rightmost nybble of theretrieved data word.

• If P4 is 0 but one or more of theparity bits P1, P2, or P3 is not 0, thena double bit error has occurred. Theprogram should probably inform theoperator and then halt.

• If P4 is not 0, a single bit error hasoccurred, which can be corrected.

To correct single bit errors, a byte chosenfrom table 3 is exclusive-ORed with the'"etrieved data byte. Parity bits P3, P2, and1 (P3 is most significant and P1 is least'oificant) determine which byte to selectn table 3. Table 4 demonstrates the cor-')n and detection process for several~nt cases.

Data Word01100111011001110110011101101111

AND Result Parity Bit01100111 P4 = 1

00100111 00100111 P3 = 001001011 01000011 P2 = 110001101 00000101 P1 = 0

Table 2: A sample calculation of parity bits for the binary data word07 7007 7 7 (hexadecimal 67). Since P4 is not 0, a single bit error has occurred,which can be detected (see text).

P3 P2 P1 Error Correction Byte0 0 0 100 0 1 800 1 0 400 1 1 081 0 0 201 0 1 041 1 0 021 1 1 01

Table 3: Single bit error correction table.

Data ANDE1E1 27E1 48E1 8D

ParityP4 = 0P3 = 0P2 = 0P1 = 0

CommentsAll parity bits 0 implies no error.

Case2: Single bit error.

Data AND Parity CommentsE3 P4 = 1 P4 not 0 implies single bitE3 27 P3 = 1 correctable error.E3 48 P2 = 1 P3P2P1 = 110E3 8D P1 = 0 Corrector from table 3 = 02

Data E3Exclusive OR 02Correct data E1

Case3: Single bit error.

Data AND Parity CommentsC1 P4 = 1 P4 not 0 implies single bitC1 27 P3 = 1 correctable error.C1 48 P2 = 0 P3P2P1 = 100C1 8D P1 = 0 Corrector from table 3 = 20

Data C1Exclusive OR 20Correct data E1

Case4: Double bit error.

Data AND Parity CommentsE2 P4 = 0 P4 equal to 0 and P3, P2, or P1E2 27 P3 = 0 not 0 implies a double bit error.E2 48 P2 = 0 Note: data in P3, P2, and P1 forE2 8D P1 = 1 a double bit error does not

imply any information aboutwhich bits are in error,

Table 4: Examples of uses of the Hamming code. For each ca~pthe transmittedinformation is hexadecimal £7. When 0.7 bit error is detected, the parity bitsP3, P2, and P7 are used to look up a correcting factor from table 3.

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Conclusion

The algorithm described here has beenprogrammed on several different micro-computers. Not many bytes of program areactually needed and the benefits are great.One need only read a large program fromtape into main memory several times torealize the utility of the approach. If yourcurrent read in software informs you ofany errors encountered, you probably muststill find the errors and correct them, as-suming you even know what the correctdata should be.

Hamming codes can be extended to cor-rect double bit errors, or to perform anypractical n bit detection and m bit cor-rection, but the extra memory costs canclimb fast. Also it is very easy to builda hardware Hamming generator and de-tector using only a few discrete integratedcircuits.

The Hamming code described here isboth practical and valuable. Manufacturersshould seriously consider incorporating thistechnique in hardware. The cassette enthu-siast should incorporate this technique i

any standardization or interchange effor •.Even the everyday experimenter will findthat the hour spent programming the algo-rithm will save many hours of frustrationin the future.-

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