Burst error-correcting code

In coding theory, burst error-correcting codes employ methods of correcting burst errors, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other.

Many codes have been designed to correct random errors. Sometimes, however, channels may introduce errors which are localized in a short interval. Such errors occur in a burst (called burst errors) because they occur in many consecutive bits. Examples of burst errors can be found extensively in storage mediums. These errors may be due to physical damage such as scratch on a disc or a stroke of lightning in case of wireless channels. They are not independent; they tend to be spatially concentrated. If one bit has an error, it is likely that the adjacent bits could also be corrupted. The methods used to correct random errors are inefficient to correct burst errors.

Definitions

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A burst of length 5

A burst of length [1]

Say a codeword   is transmitted, and it is received as   Then, the error vector   is called a burst of length   if the nonzero components of   are confined to   consecutive components. For example,   is a burst of length  

Although this definition is sufficient to describe what a burst error is, the majority of the tools developed for burst error correction rely on cyclic codes. This motivates our next definition.

A cyclic burst of length [1]

An error vector   is called a cyclic burst error of length   if its nonzero components are confined to   cyclically consecutive components. For example, the previously considered error vector  , is a cyclic burst of length  , since we consider the error starting at position   and ending at position  . Notice the indices are  -based, that is, the first element is at position  .

For the remainder of this article, we will use the term burst to refer to a cyclic burst, unless noted otherwise.

Burst description

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It is often useful to have a compact definition of a burst error, that encompasses not only its length, but also the pattern, and location of such error. We define a burst description to be a tuple   where   is the pattern of the error (that is the string of symbols beginning with the first nonzero entry in the error pattern, and ending with the last nonzero symbol), and   is the location, on the codeword, where the burst can be found.[1]

For example, the burst description of the error pattern   is  . Notice that such description is not unique, because   describes the same burst error. In general, if the number of nonzero components in   is  , then   will have   different burst descriptions each starting at a different nonzero entry of  . To remedy the issues that arise by the ambiguity of burst descriptions with the theorem below, however before doing so we need a definition first.

Definition. The number of symbols in a given error pattern   is denoted by  

Theorem (Uniqueness of burst descriptions) — Suppose   is an error vector of length   with two burst descriptions   and  . If   then the two descriptions are identical that is, their components are equivalent.[2]

Proof

Let   be the hamming weight (or the number of nonzero entries) of  . Then   has exactly   error descriptions. For   there is nothing to prove. So we assume that   and that the descriptions are not identical. We notice that each nonzero entry of   will appear in the pattern, and so, the components of   not included in the pattern will form a cyclic run of zeros, beginning after the last nonzero entry, and continuing just before the first nonzero entry of the pattern. We call the set of indices corresponding to this run as the zero run. We immediately observe that each burst description has a zero run associated with it and that each zero run is disjoint. Since we have   zero runs, and each is disjoint, we have a total of   distinct elements in all the zero runs. On the other hand we have:   This contradicts   Thus, the burst error descriptions are identical.

A corollary of the above theorem is that we cannot have two distinct burst descriptions for bursts of length  

Cyclic codes for burst error correction

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Cyclic codes are defined as follows: think of the   symbols as elements in  . Now, we can think of words as polynomials over   where the individual symbols of a word correspond to the different coefficients of the polynomial. To define a cyclic code, we pick a fixed polynomial, called generator polynomial. The codewords of this cyclic code are all the polynomials that are divisible by this generator polynomial.

Codewords are polynomials of degree  . Suppose that the generator polynomial   has degree  . Polynomials of degree   that are divisible by   result from multiplying   by polynomials of degree  . We have   such polynomials. Each one of them corresponds to a codeword. Therefore,   for cyclic codes.

Cyclic codes can detect all bursts of length up to  . We will see later that the burst error detection ability of any   code is bounded from above by  . Cyclic codes are considered optimal for burst error detection since they meet this upper bound:

Theorem (Cyclic burst correction capability) — Every cyclic code with generator polynomial of degree   can detect all bursts of length  

Proof

We need to prove that if you add a burst of length   to a codeword (i.e. to a polynomial that is divisible by  ), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by  ). It suffices to show that no burst of length   is divisible by  . Such a burst has the form  , where   Therefore,   is not divisible by   (because the latter has degree  ).   is not divisible by   (Otherwise, all codewords would start with  ). Therefore,   is not divisible by   as well.

The above proof suggests a simple algorithm for burst error detection/correction in cyclic codes: given a transmitted word (i.e. a polynomial of degree  ), compute the remainder of this word when divided by  . If the remainder is zero (i.e. if the word is divisible by  ), then it is a valid codeword. Otherwise, report an error. To correct this error, subtract this remainder from the transmitted word. The subtraction result is going to be divisible by   (i.e. it is going to be a valid codeword).

By the upper bound on burst error detection ( ), we know that a cyclic code can not detect all bursts of length  . However cyclic codes can indeed detect most bursts of length  . The reason is that detection fails only when the burst is divisible by  . Over binary alphabets, there exist   bursts of length  . Out of those, only   are divisible by  . Therefore, the detection failure probability is very small ( ) assuming a uniform distribution over all bursts of length  .

We now consider a fundamental theorem about cyclic codes that will aid in designing efficient burst-error correcting codes, by categorizing bursts into different cosets.

Theorem (Distinct Cosets) — A linear code   is an  -burst-error-correcting code if all the burst errors of length   lie in distinct cosets of  .

Proof

Let   be distinct burst errors of length   which lie in same coset of code  . Then   is a codeword. Hence, if we receive   we can decode it either to   or  . In contrast, if all the burst errors   and   do not lie in same coset, then each burst error is determined by its syndrome. The error can then be corrected through its syndrome. Thus, a linear code   is an  -burst-error-correcting code if and only if all the burst errors of length   lie in distinct cosets of  .

Theorem (Burst error codeword classification) — Let   be a linear  -burst-error-correcting code. Then no nonzero burst of length   can be a codeword.

Proof

Let   be a codeword with a burst of length  . Thus it has the pattern  , where   and   are words of length   Hence, the words   and   are two bursts of length  . For binary linear codes, they belong to the same coset. This contradicts the Distinct Cosets Theorem, therefore no nonzero burst of length   can be a codeword.

Burst error correction bounds

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Upper bounds on burst error detection and correction

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By upper bound, we mean a limit on our error detection ability that we can never go beyond. Suppose that we want to design an   code that can detect all burst errors of length   A natural question to ask is: given   and  , what is the maximum   that we can never achieve beyond? In other words, what is the upper bound on the length   of bursts that we can detect using any   code? The following theorem provides an answer to this question.

Theorem (Burst error detection ability) — The burst error detection ability of any   code is  

Proof

First we observe that a code can detect all bursts of length   if and only if no two codewords differ by a burst of length  . Suppose that we have two code words   and   that differ by a burst   of length  . Upon receiving  , we can not tell whether the transmitted word is indeed   with no transmission errors, or whether it is   with a burst error   that occurred during transmission. Now, suppose that every two codewords differ by more than a burst of length   Even if the transmitted codeword   is hit by a burst   of length  , it is not going to change into another valid codeword. Upon receiving it, we can tell that this is   with a burst   By the above observation, we know that no two codewords can share the first   symbols. The reason is that even if they differ in all the other   symbols, they are still going to be different by a burst of length   Therefore, the number of codewords   satisfies   Applying   to both sides and rearranging, we can see that  .

Now, we repeat the same question but for error correction: given   and  , what is the upper bound on the length   of bursts that we can correct using any   code? The following theorem provides a preliminary answer to this question:

Theorem (Burst error correction ability) — The burst error correction ability of any   code satisfies  

Proof

First we observe that a code can correct all bursts of length   if and only if no two codewords differ by the sum of two bursts of length   Suppose that two codewords   and   differ by bursts   and   of length   each. Upon receiving   hit by a burst  , we could interpret that as if it was   hit by a burst  . We can not tell whether the transmitted word is   or  . Now, suppose that every two codewords differ by more than two bursts of length  . Even if the transmitted codeword   is hit by a burst of length  , it is not going to look like another codeword that has been hit by another burst. For each codeword   let   denote the set of all words that differ from   by a burst of length   Notice that   includes   itself. By the above observation, we know that for two different codewords   and   and   are disjoint. We have   codewords. Therefore, we can say that  . Moreover, we have  . By plugging the latter inequality into the former, then taking the base   logarithm and rearranging, we get the above theorem.

A stronger result is given by the Rieger bound:

Theorem (Rieger bound) — If   is the burst error correcting ability of an   linear block code, then  .

Proof

Any linear code that can correct any burst pattern of length   cannot have a burst of length   as a codeword. If it had a burst of length   as a codeword, then a burst of length   could change the codeword to a burst pattern of length  , which also could be obtained by making a burst error of length   in all zero codeword. If vectors are non-zero in first   symbols, then the vectors should be from different subsets of an array so that their difference is not a codeword of bursts of length  . Ensuring this condition, the number of such subsets is at least equal to number of vectors. Thus, the number of subsets would be at least  . Hence, we have at least   distinct symbols, otherwise, the difference of two such polynomials would be a codeword that is a sum of two bursts of length   Thus, this proves the Rieger Bound.

Definition. A linear burst-error-correcting code achieving the above Rieger bound is called an optimal burst-error-correcting code.

Further bounds on burst error correction

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There is more than one upper bound on the achievable code rate of linear block codes for multiple phased-burst correction (MPBC). One such bound is constrained to a maximum correctable cyclic burst length within every subblock, or equivalently a constraint on the minimum error free length or gap within every phased-burst. This bound, when reduced to the special case of a bound for single burst correction, is the Abramson bound (a corollary of the Hamming bound for burst-error correction) when the cyclic burst length is less than half the block length.[3]

Theorem (number of bursts) — For   over a binary alphabet, there are   vectors of length   which are bursts of length  .[1]

Proof

Since the burst length is   there is a unique burst description associated with the burst. The burst can begin at any of the   positions of the pattern. Each pattern begins with   and contain a length of  . We can think of it as the set of all strings that begin with   and have length  . Thus, there are a total of   possible such patterns, and a total of   bursts of length   If we include the all-zero burst, we have   vectors representing bursts of length  

Theorem (Bound on the number of codewords) — If   a binary  -burst error correcting code has at most   codewords.

Proof

Since  , we know that there are   bursts of length  . Say the code has   codewords, then there are   codewords that differ from a codeword by a burst of length  . Each of the   words must be distinct, otherwise the code would have distance  . Therefore,   implies  

Theorem (Abramson's bounds) — If   is a binary linear  -burst error correcting code, its block-length must satisfy:  

Proof

For a linear   code, there are   codewords. By our previous result, we know that   Isolating  , we get  . Since   and   must be an integer, we have  .

Remark.   is called the redundancy of the code and in an alternative formulation for the Abramson's bounds is  

Fire codes

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Sources:[3][4][5]

While cyclic codes in general are powerful tools for detecting burst errors, we now consider a family of binary cyclic codes named Fire Codes, which possess good single burst error correction capabilities. By single burst, say of length  , we mean that all errors that a received codeword possess lie within a fixed span of   digits.

Let   be an irreducible polynomial of degree   over  , and let   be the period of  . The period of  , and indeed of any polynomial, is defined to be the least positive integer   such that   Let   be a positive integer satisfying   and   not divisible by  , where   and   are the degree and period of  , respectively. Define the Fire Code   by the following generator polynomial:  

We will show that   is an  -burst-error correcting code.

Lemma 1 —  

Proof

Let   be the greatest common divisor of the two polynomials. Since   is irreducible,   or  . Assume   then   for some constant  . But,   is a divisor of   since   is a divisor of  . But this contradicts our assumption that   does not divide   Thus,   proving the lemma.

Lemma 2 — If   is a polynomial of period  , then   if and only if  

Proof

If  , then  . Thus,  

Now suppose  . Then,  . We show that   is divisible by   by induction on  . The base case   follows. Therefore, assume  . We know that   divides both (since it has period  )   But   is irreducible, therefore it must divide both   and  ; thus, it also divides the difference of the last two polynomials,  . Then, it follows that   divides  . Finally, it also divides:  . By the induction hypothesis,  , then  .

A corollary to Lemma 2 is that since   has period  , then   divides   if and only if  .

Theorem — The Fire Code is  -burst error correcting[4][5]

If we can show that all bursts of length   or less occur in different cosets, we can use them as coset leaders that form correctable error patterns. The reason is simple: we know that each coset has a unique syndrome decoding associated with it, and if all bursts of different lengths occur in different cosets, then all have unique syndromes, facilitating error correction.

Proof of Theorem

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Let   and   be polynomials with degrees   and  , representing bursts of length   and   respectively with   The integers   represent the starting positions of the bursts, and are less than the block length of the code. For contradiction sake, assume that   and   are in the same coset. Then,   is a valid codeword (since both terms are in the same coset). Without loss of generality, pick  . By the division theorem we can write:   for integers   and  . We rewrite the polynomial   as follows:  

Notice that at the second manipulation, we introduced the term  . We are allowed to do so, since Fire Codes operate on  . By our assumption,   is a valid codeword, and thus, must be a multiple of  . As mentioned earlier, since the factors of   are relatively prime,   has to be divisible by  . Looking closely at the last expression derived for   we notice that   is divisible by   (by the corollary of Lemma 2). Therefore,   is either divisible by   or is  . Applying the division theorem again, we see that there exists a polynomial   with degree   such that:  

Then we may write:  

Equating the degree of both sides, gives us   Since   we can conclude   which implies   and  . Notice that in the expansion:   The term   appears, but since  , the resulting expression   does not contain  , therefore   and subsequently   This requires that  , and  . We can further revise our division of   by   to reflect   that is  . Substituting back into   gives us,  

Since  , we have  . But   is irreducible, therefore   and   must be relatively prime. Since   is a codeword,   must be divisible by  , as it cannot be divisible by  . Therefore,   must be a multiple of  . But it must also be a multiple of  , which implies it must be a multiple of   but that is precisely the block-length of the code. Therefore,   cannot be a multiple of   since they are both less than  . Thus, our assumption of   being a codeword is incorrect, and therefore   and   are in different cosets, with unique syndromes, and therefore correctable.

Example: 5-burst error correcting fire code

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With the theory presented in the above section, consider the construction of a  -burst error correcting Fire Code. Remember that to construct a Fire Code, we need an irreducible polynomial  , an integer  , representing the burst error correction capability of our code, and we need to satisfy the property that   is not divisible by the period of  . With these requirements in mind, consider the irreducible polynomial  , and let  . Since   is a primitive polynomial, its period is  . We confirm that   is not divisible by  . Thus,   is a Fire Code generator. We can calculate the block-length of the code by evaluating the least common multiple of   and  . In other words,  . Thus, the Fire Code above is a cyclic code capable of correcting any burst of length   or less.

Binary Reed–Solomon codes

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Certain families of codes, such as Reed–Solomon, operate on alphabet sizes larger than binary. This property awards such codes powerful burst error correction capabilities. Consider a code operating on  . Each symbol of the alphabet can be represented by   bits. If   is an   Reed–Solomon code over  , we can think of   as an   code over  .

The reason such codes are powerful for burst error correction is that each symbol is represented by   bits, and in general, it is irrelevant how many of those   bits are erroneous; whether a single bit, or all of the   bits contain errors, from a decoding perspective it is still a single symbol error. In other words, since burst errors tend to occur in clusters, there is a strong possibility of several binary errors contributing to a single symbol error.

Notice that a burst of   errors can affect at most   symbols, and a burst of   can affect at most   symbols. Then, a burst of   can affect at most   symbols; this implies that a  -symbols-error correcting code can correct a burst of length at most  .

In general, a  -error correcting Reed–Solomon code over   can correct any combination of   or fewer bursts of length  , on top of being able to correct  -random worst case errors.

An example of a binary RS code

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Let   be a   RS code over  . This code was employed by NASA in their Cassini-Huygens spacecraft.[6] It is capable of correcting   symbol errors. We now construct a Binary RS Code   from  . Each symbol will be written using   bits. Therefore, the Binary RS code will have   as its parameters. It is capable of correcting any single burst of length  .

Interleaved codes

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Interleaving is used to convert convolutional codes from random error correctors to burst error correctors. The basic idea behind the use of interleaved codes is to jumble symbols at the transmitter. This leads to randomization of bursts of received errors which are closely located and we can then apply the analysis for random channel. Thus, the main function performed by the interleaver at transmitter is to alter the input symbol sequence. At the receiver, the deinterleaver will alter the received sequence to get back the original unaltered sequence at the transmitter.

Burst error correcting capacity of interleaver

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Illustration of row- and column-major order

Theorem — If the burst error correcting ability of some code is   then the burst error correcting ability of its  -way interleave is  

Proof

Suppose that we have an   code that can correct all bursts of length   Interleaving can provide us with a   code that can correct all bursts of length   for any given  . If we want to encode a message of an arbitrary length using interleaving, first we divide it into blocks of length  . We write the   entries of each block into a   matrix using row-major order. Then, we encode each row using the   code. What we will get is a   matrix. Now, this matrix is read out and transmitted in column-major order. The trick is that if there occurs a burst of length   in the transmitted word, then each row will contain approximately   consecutive errors (More specifically, each row will contain a burst of length at least   and at most  ). If   then   and the   code can correct each row. Therefore, the interleaved   code can correct the burst of length  . Conversely, if   then at least one row will contain more than   consecutive errors, and the   code might fail to correct them. Therefore, the error correcting ability of the interleaved   code is exactly   The BEC efficiency of the interleaved code remains the same as the original   code. This is true because:  

Block interleaver

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The figure below shows a 4 by 3 interleaver.

 
An example of a block interleaver

The above interleaver is called as a block interleaver. Here, the input symbols are written sequentially in the rows and the output symbols are obtained by reading the columns sequentially. Thus, this is in the form of   array. Generally,   is length of the codeword.

Capacity of block interleaver: For an   block interleaver and burst of length   the upper limit on number of errors is   This is obvious from the fact that we are reading the output column wise and the number of rows is  . By the theorem above for error correction capacity up to   the maximum burst length allowed is   For burst length of  , the decoder may fail.

Efficiency of block interleaver ( ): It is found by taking ratio of burst length where decoder may fail to the interleaver memory. Thus, we can formulate   as  

Drawbacks of block interleaver : As it is clear from the figure, the columns are read sequentially, the receiver can interpret single row only after it receives complete message and not before that. Also, the receiver requires a considerable amount of memory in order to store the received symbols and has to store the complete message. Thus, these factors give rise to two drawbacks, one is the latency and other is the storage (fairly large amount of memory). These drawbacks can be avoided by using the convolutional interleaver described below.

Convolutional interleaver

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Cross interleaver is a kind of multiplexer-demultiplexer system. In this system, delay lines are used to progressively increase length. Delay line is basically an electronic circuit used to delay the signal by certain time duration. Let   be the number of delay lines and   be the number of symbols introduced by each delay line. Thus, the separation between consecutive inputs =   symbols. Let the length of codeword   Thus, each symbol in the input codeword will be on distinct delay line. Let a burst error of length   occur. Since the separation between consecutive symbols is   the number of errors that the deinterleaved output may contain is   By the theorem above, for error correction capacity up to  , maximum burst length allowed is   For burst length of   decoder may fail.

 
An example of a convolutional interleaver
 
An example of a deinterleaver

Efficiency of cross interleaver ( ): It is found by taking the ratio of burst length where decoder may fail to the interleaver memory. In this case, the memory of interleaver can be calculated as  

Thus, we can formulate   as follows:  

Performance of cross interleaver : As shown in the above interleaver figure, the output is nothing but the diagonal symbols generated at the end of each delay line. In this case, when the input multiplexer switch completes around half switching, we can read first row at the receiver. Thus, we need to store maximum of around half message at receiver in order to read first row. This drastically brings down the storage requirement by half. Since just half message is now required to read first row, the latency is also reduced by half which is good improvement over the block interleaver. Thus, the total interleaver memory is split between transmitter and receiver.

Applications

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Compact disc

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Without error correcting codes, digital audio would not be technically feasible.[7] The Reed–Solomon codes can correct a corrupted symbol with a single bit error just as easily as it can correct a symbol with all bits wrong. This makes the RS codes particularly suitable for correcting burst errors.[5] By far, the most common application of RS codes is in compact discs. In addition to basic error correction provided by RS codes, protection against burst errors due to scratches on the disc is provided by a cross interleaver.[3]

Current compact disc digital audio system was developed by N. V. Philips of The Netherlands and Sony Corporation of Japan (agreement signed in 1979).

A compact disc comprises a 120 mm aluminized disc coated with a clear plastic coating, with spiral track, approximately 5 km in length, which is optically scanned by a laser of wavelength ~0.8 μm, at a constant speed of ~1.25 m/s. For achieving this constant speed, rotation of the disc is varied from ~8 rev/s while scanning at the inner portion of the track to ~3.5 rev/s at the outer portion. Pits and lands are the depressions (0.12 μm deep) and flat segments constituting the binary data along the track (0.6 μm width).[8]

The CD process can be abstracted as a sequence of the following sub-processes:

  • Channel encoding of source of signals
  • Mechanical sub-processes of preparing a master disc, producing user discs and sensing the signals embedded on user discs while playing – the channel
  • Decoding the signals sensed from user discs

The process is subject to both burst errors and random errors.[7] Burst errors include those due to disc material (defects of aluminum reflecting film, poor reflective index of transparent disc material), disc production (faults during disc forming and disc cutting etc.), disc handling (scratches – generally thin, radial and orthogonal to direction of recording) and variations in play-back mechanism. Random errors include those due to jitter of reconstructed signal wave and interference in signal. CIRC (Cross-Interleaved Reed–Solomon code) is the basis for error detection and correction in the CD process. It corrects error bursts up to 3,500 bits in sequence (2.4 mm in length as seen on CD surface) and compensates for error bursts up to 12,000 bits (8.5 mm) that may be caused by minor scratches.

Encoding: Sound-waves are sampled and converted to digital form by an A/D converter. The sound wave is sampled for amplitude (at 44.1 kHz or 44,100 pairs, one each for the left and right channels of the stereo sound). The amplitude at an instance is assigned a binary string of length 16. Thus, each sample produces two binary vectors from   or 4   bytes of data. Every second of sound recorded results in 44,100 × 32 = 1,411,200 bits (176,400 bytes) of data.[5] The 1.41 Mbit/s sampled data stream passes through the error correction system eventually getting converted to a stream of 1.88 Mbit/s.

Input for the encoder consists of input frames each of 24 8-bit symbols (12 16-bit samples from the A/D converter, 6 each from left and right data (sound) sources). A frame can be represented by   where   and   are bytes from the left and right channels from the   sample of the frame.

Initially, the bytes are permuted to form new frames represented by   where   represent  -th left and right samples from the frame after 2 intervening frames.

Next, these 24 message symbols are encoded using C2 (28,24,5) Reed–Solomon code which is a shortened RS code over  . This is two-error-correcting, being of minimum distance 5. This adds 4 bytes of redundancy,   forming a new frame:  . The resulting 28-symbol codeword is passed through a (28.4) cross interleaver leading to 28 interleaved symbols. These are then passed through C1 (32,28,5) RS code, resulting in codewords of 32 coded output symbols. Further regrouping of odd numbered symbols of a codeword with even numbered symbols of the next codeword is done to break up any short bursts that may still be present after the above 4-frame delay interleaving. Thus, for every 24 input symbols there will be 32 output symbols giving  . Finally one byte of control and display information is added.[5] Each of the 33 bytes is then converted to 17 bits through EFM (eight to fourteen modulation) and addition of 3 merge bits. Therefore, the frame of six samples results in 33 bytes × 17 bits (561 bits) to which are added 24 synchronization bits and 3 merging bits yielding a total of 588 bits.

Decoding: The CD player (CIRC decoder) receives the 32 output symbol data stream. This stream passes through the decoder D1 first. It is up to individual designers of CD systems to decide on decoding methods and optimize their product performance. Being of minimum distance 5 The D1, D2 decoders can each correct a combination of   errors and   erasures such that  .[5] In most decoding solutions, D1 is designed to correct single error. And in case of more than 1 error, this decoder outputs 28 erasures. The deinterleaver at the succeeding stage distributes these erasures across 28 D2 codewords. Again in most solutions, D2 is set to deal with erasures only (a simpler and less expensive solution). If more than 4 erasures were to be encountered, 24 erasures are output by D2. Thereafter, an error concealment system attempts to interpolate (from neighboring symbols) in case of uncorrectable symbols, failing which sounds corresponding to such erroneous symbols get muted.

Performance of CIRC:[7] CIRC conceals long bust errors by simple linear interpolation. 2.5 mm of track length (4000 bits) is the maximum completely correctable burst length. 7.7 mm track length (12,300 bits) is the maximum burst length that can be interpolated. Sample interpolation rate is one every 10 hours at Bit Error Rate (BER)   and 1000 samples per minute at BER =   Undetectable error samples (clicks): less than one every 750 hours at BER =   and negligible at BER =  .

See also

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References

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  1. ^ a b c d Fong, W.H. (2011). "Coding Bounds for Multiple Phased-Burst Correction and Single Burst Correction Codes". arXiv:1104.1408 [cs.IT].
  2. ^ McEliece, R.J. (2004). The Theory of Information and Coding (Student ed.). Cambridge University Press. ISBN 978-0-521-83185-7.
  3. ^ a b c Ling, San; Xing, Chaoping (2004). Coding Theory: A First Course. Cambridge University Press. ISBN 978-0-521-52923-5.
  4. ^ a b Moon, Todd K. (2005). Error Correction Coding: Mathematical Methods and Algorithms. Wiley. ISBN 978-0-471-64800-0.
  5. ^ a b c d e f Lin, Shu; Costello, Daniel J. (2004). Error Control Coding: Fundamentals and Applications (2nd ed.). Pearson-Prentice Hall. ISBN 978-0-13-017973-9.
  6. ^ "Cassini: What kind of error correction?". quest.arc.nasa.gov. 1999. Archived from the original on 2012-06-27.
  7. ^ a b c Algebraic Error Control Codes (Autumn 2012) – Handouts from Stanford University
  8. ^ McEliece, Robert J. (1977). The Theory of Information and Coding: A Mathematical Framework for Communication. Advanced Book Program. Addison-Wesley.
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