If the code is used merely to detect errors, it can find a maximum 6 of them in any bit codeword. However, if correction is performed, only three bits are correctable. Thus, we trade identifiable errors for correctability. Keep this trade-off in mind as you examine the performance of the Golay code and the requirements of your application.
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If the code is used merely to detect errors, it can find a maximum 6 of them in any bit codeword. However, if correction is performed, only three bits are correctable. Thus, we trade identifiable errors for correctability.
Keep this trade-off in mind as you examine the performance of the Golay code and the requirements of your application. Error correction is obviously no panacea and does carry a penalty. Let me explain why. When a codeword is being corrected, the correction algorithm looks for the closest matching codeword. When we feed this codeword into the correction function, it returns 68E4E6h as the closest matching codeword, not Eh.
The receiver can use the parity bit to detect this error, but higher even numbers of errors it cannot detect. This is called an uncorrectable error, and illustrates the fact that every correction algorithm has a bit- error-rate BER limit, beyond which it cannot compensate. Fortunately, you may enable the correction facilities of the given Golay C routines according to your needs. Simplified illustration of Golay [23,12] error correction.
The diagram above illustrates what is happening during codeword correction. We see three valid Golay codewords, A and B. The X-axis represents Hamming distance between these codewords. The Y-axis represents the number of bit errors incurred as you move away from a valid codeword. Picture the correction algorithm as taking the corrupt input codeword marked in red and sliding down the slope to the nearest correct codeword.
If it is a very corrupted version of A, the correction algorithm lies to us and returns B as the corrected codeword. You can see that the minimum distance, d, of the code controls the amount of corruption that can be tolerated. This is a simplified explanation, of course. In reality, this diagram is multidimensional, and each Golay codeword has many error laden companions to keep it company in the abstract boredom of n-space. Now for some mathematical curiosities.
Among error correction codes, there is a class known as perfect codes. Briefly, perfect codes are defined as those where each of the invalid codewords, when pumped through the correction process, will be transformed into a valid codeword. There are no orphan uncorrectable information vectors. Included as perfect codes are the Hamming codes, a one-bit correction scheme, and the binary and ternary Golay codes.
The Golay codes are the only nontrivial multiple error correcting perfect codes that exist3. The perfect quality of the Golay code makes it an object of beauty in the eyes of mathematicians, especially when this property is expressed in terms of the optimum packing of spheres into a region of space4.
However, the perfection of a code is not necessarily of concern to communications engineers. The Golay codeword has some very interesting properties: Cyclic Invariance. If you take a bit Golay codeword and cyclically shift it by any number of bits, the result is also a valid Golay codeword. If you take a bit Golay codeword and invert it, the result is also a valid Golay codeword. Minimum Hamming Distance. The distance between any two Golay [23,12] codewords is always seven or more bits.
The distance between any two Golay [24,12] codewords is always eight or more bits. Error Correction. The correction algorithm can detect and correct up to three bit errors per codeword. The Golay code is obviously not able to encode a large amount of data in one codeword. Twelve bits is the maximum allowed per 23 or 24 bit codeword.
So what is its use? Well, one advantage to error correction is the elimination of communications retries which can bog down a noisy channel. In some cases, the overhead associated with a resend request is much greater than the length of a data packet. For example, to send a data packet on a half duplex or simplex radio system, the microprocessor must turn on the transmitter and wait for it come up to full power, typically up to ms. The originating station must incur the same pre- transmit delay when resending its data.
Even if all goes well, there is ms of wasted time involved in a resend request. At bits per second, ms represents bits of data, a reasonable message length in some systems.
And if the radio channel is busy with other traffic, the delays can be longer. The Golay code can reduce the number of retransmission events by allowing the receiving end to correct some errors in the received data, decreasing the probability that the channel will get overloaded.
In other situations, there may be no resend request possible, say, with a one-way RF, infrared or ultrasonic data link. The Golay code allows such systems to increase the probability of one-way, error-free, reception. Some communications channels are more prone than others to burst errors, where many consecutive data bits are corrupted. The Golay code is not alone able to correct bursts of errors over three bits long in a single codeword.
Of course, a disadvantage of the extended Golay code is that you must transmit as many check bits as data bits. Maybe you are asking, why not just send the data twice, without check bits? The answer is that no error correction is possible in such a system. How would you know which of two different messages, if not both, were corrupt? The Golay code allows error correction in exchange for the data-doubling price.
However, there are only 12 information bits per codeword, so you must break your data down into bit chunks and encode each as one codeword. Yes, there are two. You only need use one of them because they both have the same properties as mentioned above, though they generate differing checkbits. Pull a coin and choose a polynomial! We will use the coefficients of the X terms of the first polynomial, AE3h. An example worked out by hand illustrates the modulo-2 division process.
Remember that in modulo-2 division we exclusive-OR instead of subtract. The data we wish to encode is h. To generate the 11 check bits, append 11 zeros onto the bit-reversed data LSB first and perform the division below. Putting the codeword together for transmission, we get h. A parity bit could be added to the codeword to make an extended Golay code.
Of course, you can scramble the bits around in any order for transmission, just as long as they are reassembled correctly before decoding. Grouping the checkbits and information bits together as we have done here is called systematic encoding. The encoding algorithm is shown below.
Long integers are used to conveniently store one codeword. The format of the returned longint is [checkbits 11 ,data 12 ]. It is shown below. If parity is even, 0 is returned, else 1. First, we use the overall parity bit to check for odd numbers of errors, possibly failing the codeword on that basis. If parity is correct, we compute the bit syndrome of the codeword and check for zero. The syndrome is the remainder left after the codeword is divided by the generating polynomial, modulo If the codeword is a valid Golay codeword, the syndrome is zero, else non-zero.
The computation below shows a sample calculation using the previously constructed codeword, h, bit reversed for the division process. You could use a table of intermediate results to speed the process. As above, you can detect up to 6 bit errors per codeword and, with the parity bit, all odd numbers of errors. Implementation — Correcting Errors Cue the smoke and mirrors; error correction is not so trivial. It relies on the cyclic properties of Golay codewords in a scheme called systematic search decoding2.
There are other methods of Golay error correction listed in the references, though I have found this one easy to implement in software. Here is a sketch of the systematic search algorithm. Compute the syndrome of the codeword. If zero, no errors exist so go to step 5. If a trial bit has been toggled, go to step 2a, else 2. If the syndrome has a weight of three or less, the syndrome matches the error pattern bit-for-bit, and can be used to exclusive-OR the errors out of the codeword; if so, remove the errors and go to step 5, else proceed to step 3.
If the syndrome has a weight of one or two, the syndrome matches the error pattern bit-for-bit, and can be used to exclusive-OR the errors out of the codeword; if so, remove the errors and go to step 5, else proceed. Toggle a trial bit in the codeword in an effort to eliminate one bit error. Restore any previously toggled trial bit. If all 23 bits have been toggled and tried once, go to step 4; else go to step 2a. Rotate the codeword cyclically left by one bit.
Go to step 1. Rotate the codeword right to its original position, if needed. The idea is to fiddle with the codeword until the syndrome has a weight of three or less, in which case we can exclusive-OR the syndrome with the codeword to negate the errors.
However, if a trial bit has been toggled, we might have introduced an error making a total of 4 , so the threshold for exclusive-ORing the errors away in step 2a must be reduced by one for safety, to two or less.
Gall In a bit code word, change those places to 1. Start a list with the bit 0 word … The bit Golay code is called a semiperfect code. Add the first bit word that has eight or more differences from all words in the list. Details This Demonstration builds the Golay code notea four different ways. Each turn flips between one and seven coins such that the leftmost flipped coin goes from heads to tails. Sloane, Sphere Packings, Lattices, and Groups3rd ed. The Golay code is thus an error-correcting code.
GOLAY NOTES ON DIGITAL CODING PDF
If one interprets the support of each subset as a codeword of length 24 with Hamming-weight 8 , these are the "octads" in the binary Golay code. The entire Golay code can be obtained by repeatedly taking the symmetric differences of subsets, i. An easier way to write down the Steiner system resp. Winning positions in the mathematical game of Mogul: a position in Mogul is a row of 24 coins. Each turn consists of flipping from one to seven coins such that the leftmost of the flipped coins goes from head to tail. The losing positions are those with no legal move. If heads are interpreted as 1 and tails as 0 then moving to a codeword from the extended binary Golay code guarantees it will be possible to force a win.
Binary Golay code
The code words are winning positions in the game of Mogul, played with 24 coins in a row. Marcel J. Golay — Wikipedia The automorphism group is the Mathieu group The code words of weight eight are elements of an 5, 8, 24 Steiner system. Today, this paper is considered one of the most remarkable papers ever published, with deep, deep connections to group theory, graph theory, number theory, combinatorics, game theory, multidimensional geometry, and even particle physics.
Using The Golay Error Detection And Correction Code