Error correction code processor employing adjustable correction power for miscorrection minimization

[0002] This application is related to U.S. Pat. No. 5,446,743 entitled “COEFFICIENT UPDATING METHOD AND APPARATUS FOR REED-SOLOMON DECODER,” and U.S. Pat. No. 5,467,297 entitled “FINITE FIELD INVERSION,” which are incorporated herein by reference.

[0003] The present invention relates to error correcting systems for computer storage devices, particularly to an enhanced method and apparatus for correcting a multi-dimensional code, such as a product code, by adjusting the correction power during the iterative passes to minimize miscorrections.

[0004] In computer storage devices (such as magnetic and optical disc drives) the bandwidth of the recording channel is limited, as well is the signal power. To achieve performance gains, various coding techniques are employed to increase the effective signal-to-noise ratio (SNR) by increasing the system's immunity to noise. This allows an increase in storage capacity by increasing the recording density while maintaining an arbitrarily low bit error rate.

[0005] In error correction coding (ECC), the binary data to be recorded are processed mathematically to generate redundancy symbols that are appended to the data to form codewords which are written to the disc storage medium. Upon readback, the recorded codewords are estimated (detected) from the read signal, and the redundancy symbols are used to decode the estimated codewords back into the originally recorded user data. The redundancy symbols provide, in effect, a buffer which shields the codeword from noise as it passes through the recording channel. When enough noise “penetrates” this buffer, it mutates a written codeword into a different received codeword, thereby resulting in an error when decoded into the user data.

[0006] The more redundancy symbols employed in an error correction code, the larger the buffer around the codeword and the more noise that can be tolerated before a decoding error occurs. However, there is an upper bound on the performance of any given recording channel known as the “channel capacity” which refers to the maximum user data transmission rate (or recording density) achievable for a given channel while maintaining an arbitrarily low bit error rate. Ultimately, the channel capacity is a function of the channel bandwidth and the signal to noise ratio (SNR). As mentioned above, error correction codes are a means for improving performance by increasing the effective SNR.

[0007] There are many approaches to encoding/decoding the user data in order to maximize the reliability and efficiency of a recording channel; ultimately, the goal is to design a system that approaches the channel capacity while minimizing the implementation complexity and cost. Block error correcting codes are commonly employed in disc storage systems, particularly the Reed-Solomon block code due to its excellent error correction properties and low implementation cost and complexity.

[0008] Block codes encode a k-symbol input block of the source data stream into an n-symbol output block or codeword where n-k is the number of redundancy symbols and k/n is referred to as the code rate. The codewords are then transmitted through (stored to) the communication medium and decoded by the receiver. The encoding process performs a mathematical operation over the input block such that the output codewords are different from one another by a parameter referred to as the minimum distance of the code d_min. The minimum distance d_minbetween codewords determines the amount of noise that the system can tolerate before a received codeword is decoded erroneously.

[0009] With Reed-Solomon codes, the data stream is processed as a sequence of symbols, where the symbols are typically selected from a finite field GF(2^w). The parameter w denotes the number of binary data bits per symbol. Each symbol of the k-symbol input block represents the coefficients of a data polynomial D(x). The redundancy symbols (which are also represented as a polynomial W(x)) are then computed as the modulo division of the input data polynomial D(x) divided by a generator polynomial G(x):

[0010] W(x)=(x^m·D(x))MOD G(x)

[0011] where m is the degree of the generator polynomial which equals the number of redundancy symbols. The redundancy polynomial W(x) is then added to the data polynomial D(x) to generate a codeword polynomial C(x):

[0012] C(x)=(x^m·D(x))+W(x).

[0013] Those skilled in the art understand that the encoder circuitry for performing the above operations can be implemented with minimum cost using a linear feedback shift register (LFSR).

[0014] After encoding, the codeword C(x) is transmitted through the noisy communication channel, wherein the received codeword C′(x) equals the transmitted codeword C(x) plus an error polynomial E(x). The received codeword C′(x) is corrected according to the following steps: (1) compute error syndromes S_i; (2) compute the coefficients of an error locator polynomial using the error syndromes S_i; (3) compute the roots of the error locator polynomial, the logs of the roots are the error locations l_i; and (4) compute the error values using the error syndromes S_iand the roots of the error locator polynomial.

[0015] The error syndromes S_iare computed as the modulo division of the received codeword polynomial C′(x) divided by the factors of the generator polynomial G(x):

[0016] S_i=C′(x)MOD(x+αⁱ)

[0017] when $G\left(x\right)=\prod _{i=0}^{m-1}\left(x+{\alpha }^i\right)$

[0018] where α is a primitive element of the finite field GF(2^w). Techniques for performing the other steps of the decoding process, computing the error locator polynomial, computing the roots of the error locator polynomial, and computing the error values, are well known by those skilled in the art. See, for example, the above referenced U.S. Pat. No. 5,446,743 entitled “COEFFICIENT UPDATING METHOD AND APPARATUS FOR REED-SOLOMON DECODER.”

[0019] The use of erasure pointers is a well known technique for increasing the power of an error correction code. An erasure pointer is an address pointing to a symbol location of a codeword where an error is likely to have occurred. For example, an erasure pointer may be generated by detecting timing phase errors or marginal amplitude in the read signal during read back. An address Ls generated to point to the codeword symbol where the timing error or amplitude attenuation was excessive. The erasure pointers are then used to augment the error correction procedure by providing information in addition to the error syndromes S_i. An erasure polynomial is generated using the erasure pointers, where each root of the erasure polynomial replaces a root in the error locator polynomial. If the number of erasure pointers equals the number of error syndromes, then the erasure polynomial replaces the error locator polynomial. Since the erasure pointer corresponds to the error location, only an error value need be computed for each erasure pointer, which accounts for the increase in the correction capability of the code. Without erasure pointers the code is capable of correcting only INT(m/2) codeword symbols where m is the degree of the generator polynomial (and the number of error syndromes), whereas with erasure pointers the code is capable of correcting n+INT((m−n)/2) codeword symbols where n is the number of erasure His pointers (in effect doubling the correction capability when the number of erasure pointers equals the number of error syndromes, i.e., when n equals m).

[0020] Another technique known in the prior art to further increase the power of an error correction code is to arrange the codewords into what is known as a multi-dimensional code, such as a two-dimensional product code. Digital Video Disc (DVD) storage systems, for example, commonly employ a two-dimensional product code shown in FIG. 2. The codewords are arranged into intersecting horizontal (row or Q) and vertical (column or P) codewords and the decoding process is carried out in iterative passes. First a pass over the horizontal codewords is performed to correct as many errors as possible; any uncorrectable horizontal codewords are left unmodified. Then a pass is made over the vertical codewords to correct as many errors as possible, where a symbol corrected in a vertical codeword also corrects the corresponding symbol for the intersecting horizontal codeword. Consequently, the horizontal codeword may be correctable during the next horizontal pass. Similarly, a symbol corrected during a horizontal pass may render a previously uncorrectable vertical codeword correctable during the next vertical pass. This iterative process continues until the entire product code is corrected, or deemed uncorrectable.

[0021] A known problem with processing product codes is that if a miscorrection occurs, it may take several iterations to recover (i.e., to correct the miscorrection), and in some cases the system may not be able to recover at all. The probability of not detecting a miscorrection is approximately 2^−kwhere k is the number of unused redundancy bits. Assuming 8-bit symbols, k equals

[0022] 8·(T−2e)

[0023] where T is the total number of redundancy symbols in a codeword and e is the number of codeword symbols that are corrected. Thus, the more codeword symbols corrected, the higher the likelihood of Ea miscorrection since there are less remnant redundancy bits to verify the corrections.

[0024] There is, therefore, a need for an enhanced error correction code (ECC) processor which minimizes the probability of miscorrecting an ECC codeword in a multi-dimensional code, such as a product code. A further aspect of the present invention is to generate an erasure polynomial concurrent with correcting the ECC codewords in one dimension, wherein the erasure polynomial is used to correct the ECC codewords in another dimension. Yet another aspect of the present invention is to generate the erasure polynomial using bit-serial techniques in order to minimize the circuit complexity and cost.

[0025] An error correction code (ECC) processor is disclosed which minimizes the probability of miscorrecting an ECC codeword in a multi-dimensional code, such as a product code, thereby avoiding the added latency in recovering from the miscorrection. Initially, the ECC processor sets the error correction power of the code to a minimum value so that the probability of making a miscorrection is low even though the probability of detecting an uncorrectable codeword is high. The ECC processor then performs iterative error correction passes over the multi-dimensional code and incrementally increases the error correction power if no corrections are male during a current pass. Increasing the correction power may render a previously uncorrectable codeword correctable, and after correcting the codeword, it may render a corresponding intersecting codeword in the other dimension correctable. If a codeword is corrected during a current pass, the correction power is reset to the minimum value for the next pass and the correction process continues. In this manner, the probability of making a miscorrection is minimized since the number of remnant redundancy symbols used to check for a miscorrection is maximized.

[0026] The above and other aspects and advantages of the present invention will be better understood by reading the following detailed description of the invention in view of the drawings, wherein:

[0027] FIG. 1 is a block diagram of the error correction code processor of the present invention, wherein the P/Q decoder comprises an error it limit controller for incrementally adjusting the correction power of the code in order to minimize miscorrections.

[0028] FIG. 2 shows the format of a two-dimensional product code typically employed in DVD optical storage devices comprising sixteen data sectors.

[0029] FIG. 3A shows details of the error corrector of FIG. 1, including an erasure polynomial generator for generating an erasure polynomial corresponding to uncorrectable codewords in a first dimension, the erasure polynomial for correcting errors in codewords in a second dimension.

[0030] FIG. 3B shows details of the error limit controller of FIG. 1, including a controller for incrementally adjusting a correction limit which is compared to a number of errors detected in a codeword from the error syndromes S_i.

[0031] FIGS. 4A and 4B show further details of the erasure polynomial generator of the present invention.

[0032] FIG. 5A is a flow diagram executed during a horizontal pass over the two-dimensional code of FIG. 2, including the steps of adjusting the correction limit and updating the erasure polynomial.

[0033] FIG. 5B is a flow diagram executed during a vertical pass over the two-dimensional code of FIG. 2, including the steps of adjusting the correction limit and updating the erasure polynomial.

[0034] FIG. 5C is a flow diagram for adding a factor to the erasure polynomial when an uncorrectable row codeword is encountered during a horizontal pass.

[0035] FIG. 5D is a flow diagram for adding a factor to the erasure polynomial when an uncorrectable column codeword is encountered during a vertical pass.

[0036] FIG. 5E is a flow diagram for adjusting the correction limit for the vertical codewords after a horizontal error correction pass.

[0037] FIG. 5F is a flow diagram for adjusting the correction limit for the horizontal codewords after a vertical error correction pass.

[0038] The data format of a two-dimensional product code typically employed in a DVD storage device is shown in FIG. 2. The product code comprises 16 data sectors, where each data sector comprises 12 horizontal codewords (Q codewords). Each horizontal codeword comprises 10 ECC redundancy symbols preferably generated according to a Reed-Solomon code. There are 182 vertical codewords (P codewords) each comprising 16 redundancy symbols as shown. The ECC redundancy symbols also form ECC codewords; that is, the ECC redundancy symbols are correctable in the same manner as the user data. Thus, there is a total of 182 vertical codewords including the 10 vertical ECC codewords on the right side, and 208 horizontal codewords including 16 horizontal ECC codewords on the bottom. At the end of each of the 16 data sectors are four CRC symbols for use in verifying the validity and completeness of the corrections made to the codewords using the ECC redundancy symbols.

[0039] An overview of the error correction code (ECC) processor of the present invention for use in a CD/DVD optical disc storage system is shown in FIG. 1. During a write operation (assuming the device is not read only) user data received from a host system are stored in a data buffer 1. A CRC generator-and-correction validator 2 then reads the user data from the buffer 1 over line 3, generates CRC redundancy symbols for use in verifying the validity of any corrections made during read back, and restores the user data with appended redundancy symbols back into the data buffer 1. Thereafter the data is again read from the data buffer 1 (including the CRIC redundancy), randomized by a data randomizer 4, and the randomized data restored to the data buffer 1. A P/Q encoder/decoder 5 then reads the randomized data from the data buffer 1, and an ECC/syndrome generator 12 generates the ECC redundancy symbols for the column (P) and row (Q) codewords to form the two-dimensional product code shown in FIG. 2. The individual P and Q codewords are restored to the data buffer 1 after appending the ECC redundancy symbols. Once the entire product code has been generated, it is read from the data buffer 1 and written to the optical storage medium 6.

[0040] If the system is configured for a compact disc (CD) data format, then additional redundancy, referred to as C1 and C2, are generated and appended to the data before writing it to the disc. Thus to facilitate the CD recording format, the error correction system comprises a C1 encoder/decoder 7, a C2 encoder/decoder 8, and an interleaver/deinterleaver 9 for implementing the well known Cross Interleave Reed-Solomon Code (CIRC). Typically a static RAM (SRAM) is employed to implement the CIRC coding process; SRAM is much faster than dynamic RAM (DRAM), the latter being used to implement the data buffer 1.

[0041] During a read operation, the process is run in reverse. If configured for CD format, then the C1 and C2 decoders make preliminary corrections to the randomized data as it is read from the optical disc 6 and stored in the data buffer 1. Once a complete product code is available in the data buffer 1, the P/Q decoder 5 begins the iterative passes over the P and Q codewords to make further corrections. The ECC/syndrome generator 12 generates ECC syndromes S_itransmitted over line 13 to an error corrector 14. The error corrector 14 uses the ECC syndromes S_ito correct errors in the individual codewords as described above. The error corrector 14 is enabled over line 16 by an error limit controller 18 which incrementally increases the correction power of the code by adjusting the maximum allowable number of codeword symbols that can be corrected during a horizontal or vertical pass. If at the end of a P or Q pass all of the ECC error syndromes are zero, indicating that the product code is error free (unless miscorrected), then the randomized data is read from the data buffer 1 and derandomized by derandomizer 4. As the data is derandomized, it is processed by the CRC generator-and-correction validator 2 to generate the CRFC syndrome. If the CRC syndrome is zero, indicating that the corrections to the P and Q codewords are valid and complete, then the data is again read from the data buffer 1, derandomized, and the derandomized data transferred to the host system. If the CIRC syndrome is non-zero, indicating that the product code has been miscorrected, then the CRC generator-and-correction validator 2 sends an error message to the host system over line 11 wherein the host system will initiate a retry operation (i.e., attempt to reread the data from the disc).

[0042] Details of the error corrector 14 of the P/Q decoder 5 of FIG. 1 are shown in FIG. 3A. The error corrector 14 receives error syndromes S_iover line 13 that are computed for each ECC codeword (horizontal and vertical). As described above, the error syndromes S_iare computed as the modulo division of the received codeword polynomial C′(x) divided by the factors of the generator polynomial G(x):

[0043] S_i=C′(x)MOD(x+αⁱ)

[0044] when $G\left(x\right)=\prod _{i=0}^{m-1}\left(x+{\alpha }^i\right)$

[0045] where α is a primitive element of the finite field GF(2^w) (w denotes the number of binary data bits per codeword symbol) and m is the degree of the generator polynomial.

[0046] The error syndromes are processed by a conventional error-locator polynomial generator 17 to generate a conventional error-locator polynomial σ(x). Methods for generating the error-locator polynomial are well known in the art—any of the known methods can be used in the present invention. An example method is disclosed in the above referenced U.S. Pat. No. 5,446,743 entitled “COEFFICIENT UPDATING METHOD AND APPARATUS FOR REED-SOLOMON DECODER.”

[0047] The error corrector of the present invention further employs an erasure polynomial generator 19 for generating an erasure polynomial σ(x)_EP. As described in greater detail below, the roots of the erasure polynomial σ(x)_EPcorrespond to uncorrectable codewords encountered in the previous ECC pass. For example, if the P/Q decoder 5 is processing the column codewords during a vertical pass, then the roots of the erasure polynomial σ(x)_EPcorrespond to the uncorrectable row codewords encountered during the previous horizontal pass. If the number of uncorrectable codewords encountered during the previous pass is less than or equal to the Aft number of ECC redundancy symbols of a codeword in the current pass, then the erasure polynomial σ(x)_EPis used in place of the conventional error-location polynomial σ(x) in order to locate and correct errors in the codewords of the current pass. Thus, the error corrector 14 of FIG. 3A comprises a selector (shown as a multiplexor 20) for selecting between the error-locator polynomial α(x) and the erasure polynomial σ(x)_EPdepending on the number of uncorrectable codewords encountered during the previous pass.

[0048] A root search and error magnitude generator 21 processes the selected polynomial to determine the error locations and error values in correcting the current ECC codeword of the current pass. The error locations are the log of the roots α^−lof the error-locator or erasure polynomial (i.e., l), and the error values are determined from the coefficients σ_iof the error-locator or erasure polynomial, the roots α^−lof that polynomial, and the error syndromes S_iusing any of the methods known in the art. For example, in the above referenced U.S. Pat. No. 5,446,743 entitled “COEFFICIENT UPDATING METHOD AND APPARATUS FOR REED-SOLOMON DECODER,” an error evaluator polynomial ω(x) is first generated by computing its coefficients according to ${\omega }_i=\sum _{j=0}^i{\sigma }_j{S}_{i-j}$

[0049] for i=0, 1, . . . , t−1, where σ_jare the coefficients of either the error-locator polynomial σ(x) or the erasure polynomial σ(x)_EP. Then the error values E_lare computed according to $E_l=\frac{\omega \left({\alpha }^{-l}\right)}{{\sigma }^{\prime }\left({\alpha }^{-l}\right)}$

[0050] where σ′ represents the first derivative of σ, and l is the error location. To correct an error, the symbol from the codeword corresponding to the error location l is read from the data buffer 1 and stored in a register 22. The correction value E_lis then added (XORed) 24 to the symbol to correct it, and the corrected symbol is restored to the data buffer 1.

[0051] If the erasure polynomial σ(x)_EPis selected to determine the error locations and error values, then it will be the last pass over the data unless miscorrections occur. It will be the last pass because the maximum number of errors in any codeword will be less than or equal to the number of redundancy symbols in the codeword. Furthermore, the erasure polynomial σ(x)_EPgenerated during the previous pass is advantageously used to correct all of the codewords of the current pass rather than regenerating it for each codeword as with the error-locator polynomial σ(x). In addition, the erasure polynomial σ(x)_EPis generated concurrent with correcting the codewords (generating the error syndromes, error-locator polynomial, etc.), which reduces the complexity and cost of the circuitry.

[0052] To reduce the probability of a miscorrecting an ECC codewords (horizontal or vertical), a correction power of the code is adjusted incrementally by the error limit controller 18 of FIG. 1. The error corrector 14 of FIG. 3A is enabled to correct an ECC codeword only if the detected number of errors is less than a limit value as explained in the following section.

[0053] The purpose of the error limit controller 18 of FIG. 1 is to adjust the correction power of the code in order to minimize the probability of miscorrecting an ECC codeword. As explained above, if a miscorrection occurs, it may take several error correction passes to recover (i.e., to correct the miscorrection) and in some cases the system may not be able to recover at all. A miscorrection will occur if there are not enough remnant redundancy symbols after making a correction to verify that the corrections are valid. Therefore, the error limit controller 18 limits the number of allowable corrections in order to maximize the number of remnant redundancy symbols used to verify the corrections. Initially, the error limit is set to a minimum value relative to the number of redundancy symbols in a horizontal or vertical codeword. If the number of detected errors in a codeword exceeds the limit value, then the codeword is marked as uncorrectable and a corresponding root is added to the erasure polynomial.

[0054] Since the limit value is set low initially, a number of codewords will be marked as uncorrectable when in fact they could be corrected. However, there may also be a number of codewords that would be miscorrected but are skipped because the number of detected errors exceeds the correction limit. Errors in what would have been miscorrected codewords may be corrected in subsequent passes, thereby avoiding the delay caused by a miscorrection. For example, if a horizontal codeword that would otherwise be miscorrected is ad skipped during the first horizontal pass, enough errors in that codeword may be corrected in a vertical pass to render it correctable in a subsequent horizontal pass. Although the present invention adds some delay by skipping codewords that could be corrected, there is a significant performance gain on average in avoiding miscorrections.

[0055] Details of the circuitry employed in the error limit controller 18 are shown in FIG. 3B. The error syndromes S_i13 are processed by a UNC detector 23 which determines whether the number and location of errors detected in a codeword are correctable irrespective of the error limit. If not, then the codeword is marked as uncorrectable UNC 27 through OR gate 25, and the error corrector 14 of FIG. 1 is disabled over control line 16 by inverting 29 the UNC signal 27. FIG. 3B shows that the error limit controller 18 also comprises a NUM ERRS detector 31 for detecting a number of errors in the current codeword from the error syndromes S_i13. In the preferred embodiment, the same circuitry in the UNC detector 23 is used to implement the NUC ERRS detector 13; they are illustrated as separate circuits in FIG. 3B to emphasize that the UNC detector 23 also checks whether the locations of the detected errors are within the bounds of the codeword (a codeword may be a subset of the entire finite field and errors can be detected outside the codeword). Implementation details for implementing the UNC detector 23 (and the NUM ERRS detector 31) are not shown because they are well known in the art; the details are not necessary to understand the present invention.

[0056] The enabling component of the present invention is the controller 33 of FIG. 3B which adjusts an error limit 35 for both the horizontal and vertical codewords based on the current error limit, the number of uncorrectable codewords detected in the previous error correction pass, and whether a correction was made to a codeword in the previous error correction pass. The number of errors detected 31 in the current codeword is compared to the error limit 35 at comparator 37; if the number of detected errors exceeds the error limit, then the codeword is marked uncorrectable 27 through OR gate 25 and the error corrector 14 is disabled over control line 16.

[0057] The controller 33 increases the error limit 35 when no other corrections can be made using the current error limit value. No other corrections can be made using the current error limit value if no corrections were made in the previous error correction pass. Thus, the controller 33 monitors the UNC line 27 and the number of errors detected 31 to determine if a correction was made during the previous pass (i.e., a correction was made if the number of detected errors 31 is greater than zero and the UNC line 27 is FALSE). If not, the controller 33 increments the error limit 35 to allow more codewords to be corrected during the next pass. If a correction is made during the previous pass, then the controller 33 resets the error limit 35 to its initial minimum value so that the probability of a miscorrection is again minimized.

[0058] If the number of uncorrectable codewords encountered during the previous pass (as counted by a UNC counter 45 in FIG. 3B) is less than or equal to the number of ECC redundancy symbols of a codeword in the current pass, then the erasure polynomial σ(x)_EPis used in place of the conventional error-location polynomial σ(x) in order to locate and correct errors in the codewords of the current pass. Accordingly, the controller 33 sets the error limit 35 to the maximum value (to the number of ECC redundancy symbols in the codewords of the current pass) because the likelihood of making a miscorrection when using the erasure polynomial is very low, therefore the error corrector 14 is allowed to correct all of the errors that correspond to the erasure pointers.

[0059] Details of the erasure polynomial generator 18 of FIG. 3 are shown in FIGS. 4A and 4B, with an operational description provided in the flow diagrams of FIGS. 5A-5F. The circuitry of FIG. 4A generates an erasure polynomial σ(x)_EPaccording to ${\sigma \left(x\right)}_{\mathrm{EP}}=\prod _i\left(1+a^{l_i}x\right)$

[0060] where l_iare the locations in the codeword where a symbol may be in error, which correspond to the uncorrectable codewords of a previous pass. Referring to FIG. 2, for example, each uncorrectable row codeword encountered during a horizontal pass corresponds to a symbol in the column codewords that may be in error. Thus, if the third row codeword is uncorrectable during a horizontal pass, then an erasure pointer is generated for the third symbol in every column codeword. Only one erasure polynomial σ(x)_EPneeds to be generated because it is the same for every column codeword (i.e., an uncorrectable row codeword affects the same symbol location in every column codeword).

[0061] The circuitry of FIG. 4A is configured such that the initial value for the erasure polynomial is σ(x)_EP=1. Then for each uncorrectable codeword encountered, the erasure polynomial σ(x)_EPis multiplied by a new factor (1+α^l) where l corresponds to the location of the uncorrectable codeword in the current pass with respect to the codewords in the next pass. Mathematically, the circuitry computes

[0062] σ(x)_EP(k+1)=σ(x)_EP(k)(1+αx)

[0063] where σ(x)_EP(k)is the previous erasure polynomial and σ(x)_EP(k+1)is the new erasure polynomial.

[0064] At this point, it is important to understand that in the circuitry of FIG. 4A the coefficients σ_iof the erasure polynomial σ(x)_EPare represented in a dual basis as opposed to the conventional basis. The reason for using a dual basis representation is to simplify the multiplier circuitry of FIG. 4A, as well as to facilitate the root search and error magnitude generator 21 of FIG. 3A which expects to receive the coefficients σ_iof the erasure polynomial σ(x)_EPin the dual basis. Details of the root search and error magnitude generator 21 and its operation in the dual basis is understood from the above referenced U.S. Pat. No. 5,446,743 entitled “COEFFICIENT UPDATING METHOD AND APPARATUS FOR REED-SOLOMON DECODER,” and U.S. Pat. No. 5,467,297 entitled “FINITE FIELD INVERSION.” As described in those patents, performing operations in the dual basis reduces the cost and complexity of the circuitry by exploiting bit-serial techniques, an aspect also employed in the present invention. To understand the present invention, it is only necessary to understand the concept of a dual basis and how it is used in the circuitry of FIG. 4A.

[0065] In a finite field GF(p^w), each element may be viewed as a vector in an w-dimensional vector space. The standard or canonical basis set is {α^k} for k=0, 1, . . . , w−1. Let {α_k} be a set of basis vectors such that α_k=α₀α^kfor k=0,1, . . . , w−1. The so called dual basis to this set is {β_k} where β_kis chosen such that tr(α_kβ_j)=α⁰if k is equal to j and tr(α_kβ_j)=0 if k is not equal to j. The trace function tr(x) is defined as $\mathrm{tr}\left(x\right)=\sum _{i=0}^{w-1}{x}^{p^i}$

[0066] Let A be an element of GF(p^w) expressed in the canonical representation which will be referred to as the a representation and let B be an element of GF(p^w) expressed in the dual basis representation which will be referred to as the β representation, that is $A=\sum _{k=0}^{w-1}{A}_k{\alpha }^k\mathrm{and}B=\sum _{k=0}^{w-1}{B}_k{\beta }_k$

[0067] where A_kare the components of the vector A and B_kare the components of the vector B. Then $\mathrm{tr}\left({\alpha }_0{\alpha }^{k}B\right)=\mathrm{tr}\left({\alpha }_k\sum _{j=0}^{w-1}{B}_j{\beta }_j\right)=\mathrm{tr}\left(\sum _{j=0}^{w-1}{B}_j{\alpha }_k{\beta }_j\right)=B_k{\alpha }^0,k=0,1,\dots ,w-1.$

[0068] Let C be the β representation of AB and let (α^kA)_ibe the components of α^kA in α representation. Then from the above equation $\begin{array}{c}{C}_k{\alpha }^0=\mathrm{tr}\left({\alpha }_0{\alpha }^k\mathrm{AB}\right),k=0,1,\dots ,w-1\\ =\mathrm{tr}\left({\alpha }_0\sum _{i=0}^{w-1}{\left({\alpha }^{k}A\right)}_i{\alpha }^i\left(\sum _{j=0}^{w-1}{B}_j{\beta }_j\right)\right),k=0,1,\dots ,w-1\\ =\sum _{i=0}^{w-1}{\left({\alpha }^{k}A\right)}_i\left(\sum _{j=0}^{w-1}{B}_j\mathrm{tr}\left({\alpha }_i{\beta }_j\right)\right),k=0,1,\dots ,w-1\\ =\sum _{i=0}^{w-1}{\left({\alpha }^{k}A\right)}_i{B}_i{\alpha }^0,k=0,1,\dots ,w-1\\ \mathrm{Thus}\\ C_k=\sum _{i=0}^{w-1}{\left({\alpha }^{k}A\right)}_i{B}_i,k=0,1,\dots ,w-1\end{array}$

[0069] that is, the kth component of the β representation of the product of A and B is the inner product of the a representation of α^kA and the β representation of B.

[0070] This concept of computing the product of two field elements in different basis representations is exploited in the present invention. As described above, to add a new factor (erasure pointer) to the erasure polynomial σ(x)_EPthe circuitry of FIG. 4A computes

[0071] σ(x)_EP(k+1)=σ(x)_EP(k+1)(1+α^lx)=σ(x)_EP(k+1)+α^lσ(x)_EP(k+1)X

[0072] where σ(x)_EP(k+1)is the previous erasure polynomial and σ(x)_EP(k+1)is the new erasure polynomial. With α^lin the α representation and the coefficients σ_iof the erasure polynomial in the β representation, then according to the above equations the circuitry of FIG. 4A computes $\begin{array}{c}{\left({\alpha }^l{\sigma }_i\right)}_k=\sum _{j=0}^{w-1}{\left({\alpha }^k{\alpha }^l\right)}_j{\left({\sigma }_i\right)}_j,k=0,1,\dots ,w-1\\ =\sum _{j=0}^{w-1}{{\alpha }_j^{l+k}\left({\sigma }_i\right)}_j,k=0,1,\dots ,w-1\end{array}$

[0073] and then adds the result of this inner product to the kth component of σ_i+1for i=15 . . . 1 (adding the inner product to the kth component of σ_i+1is easily verified from the above equation σ(x)_EP(k+1)+α^lσ(x)_EP(k+1)x).

[0074] Having derived the mathematical operation, the structure and interconnections of FIG. 4A will now be described. The circuitry comprises sixteen single-bit shift registers 26₁-26₁₆, where shift registers 26₁-26₁₆will contain the coefficients σ_1-16of the erasure polynomial σ(x)_EPfor the column codewords at the end of a horizontal pass, and shift registers 26₁-26₉and 26_15-16will contain the coefficients σ_1-10of the erasure polynomial σ(x)_EPfor the row codewords at the end of a vertical pass. The zero coefficient σ₀is always 1, therefore a register is not needed to store its value. The shift registers 26₁-26₁₆are connected in a ring over a single bit bus as shown, where a multiplexer 29 connects the output of register σ₁₄or σ₈to the input of register σ₁₅depending on whether the current pass is horizontal or vertical (i.e., the erasure polynomial σ(x)_EPcomprises sixteen coefficients for the vertical codewords and ten for the horizontal codewords corresponding to the respective number of redundancy symbols as shown in FIG. 2).

[0075] The circuitry of FIG. 4A also comprises an inner product (IP) multiplier 28 for multiplying the current coefficient stored in register 26₁₅by a PVAL value stored in register 30. The IP multiplier 28, as shown in FIG. 4B, comprises a number of AND gates 32₀-32₇and a summing XOR gate 34 to implement the sum of products in the above equation $\sum _{j=0}^{w-1}{{\alpha }_j^{l+k}\left({\sigma }_i\right)}_j$

[0076] The content of the PVAL register 30 is multiplied by an α multiplier 36 after each inner product multiplication (i.e., for k=0 . . . w−1) to implement the α^l+koperation of the above equation. An α⁻¹multiplier 38 is then used to re-adjust the α operand from α^l+8to α^l, and to decrement l before processing the next codeword.

[0077] In the embodiment shown in FIG. 4A, the symbols size of the codeword is 8 bits (i.e., w=8). Thus to add an erasure pointer to the erasure polynomial, the circuitry of FIG. 4A executes an “A phase” in 8 clock cycles to compute the inner product for the 8 bits of (α^lσ_i)_k, for k=0 . . . 7. During each clock cycle of the “A phase”, a bit is shifted out of the σ₁₆register 26₁₆and added at adder 37 to the output of the inner product multiplier 28 selected through multiplexer 39. The output of adder 37 is then restored to the σ₁₆register 26₁₆through multiplexer 41. Also during each cycle of the “A phase”, the output of the PVAL register is multiplied by α 36 and the result restored to the PVAL register 30 through multiplexer 43.

[0078] When the “A phase” finishes after 8 clock cycles, a “B phase” is executed during the next 8 clock cycles wherein the contents of the vi registers 26_1-16are shifted to the right such that σ_i+1=σ_iand σ₁=σ₁₆. Also during each clock cycle of the “B phase”, the PVAL register 30 is multiplied by α⁻¹to restore it to its original value.

[0079] The “A phase” and the “B phase” are re-iterated 16 times during a horizontal pass to update the 16 coefficients σ_1-16of the erasure polynomial used to correct the vertical codewords, or 10 times during a vertical pass to update the 10 coefficients σ_1-10of the erasure polynomial used to correct the horizontal codewords. Since the zero coefficient σ₀is always 1, during the last of the 16 or 10 iterations bit 7 of the PVAL register is selected through multiplexer 39 and added to the bits of σ₁₆(i.e., the inner product of σ₀=1=1000000 and PVAL is bit 7 of PVAL). In this manner, it is not necessary to store the σ₀coefficient in a separate register.

[0080] The operation of the ECC processor of the present invention is further understood with reference to the flow diagrams of FIGS. 5A-5F. Before the first horizontal pass, a variable VERT. ERRS is initialized to 11 at step 40 of FIG. 5A so that the erasure polynomial σ(x)_EPwill not be used to generate the error locations and error values during the first horizontal pass (the erasure polynomial is undefined during the first horizontal pass). Also at step 40, the erasure polynomial σ(x)_EPis initialized to 1 (the circuitry of FIG. 4A actually operates as if the erasure polynomial σ(x)_EPwere initially 1, as described above), and variables CHANGE_H and CHANG_V which indicate whether a correction was made during the horizontal and vertical pass, respectively, are initialized to TRUE so that the error limits will be set to their minimum values. Before each horizontal pass, the error limit for the horizontal codewords is updated at step 47 by executing the flow chart of FIG. 5E.

[0081] Referring to FIG. 5E, at step 42 certain variables are initialized for the first and subsequent horizontal passes: a counter variable HORZ. ERRS is initialized to zero to indicate that no uncorrectable row codewords have been encountered, an index variable ROW is initialized to zero to point to the first row codeword, and a PVAL register (see FIG. 4A) is initialized to α²⁰⁷corresponding to erasure location l=207 in a vertical codeword. If at step 49 the number of uncorrectable codewords encountered in the previous vertical pass is not greater than 10, then the erasure polynomial is used to correct the horizontal codewords and the error limit H_LIM for the horizontal codewords is set to 10 at step 51. Otherwise, at step 53 a branch is executed depending on whether a correction was made during the previous vertical pass. If a correction was made (CHANGE_V==TRUE) or it is the first horizontal pass, then at step 55 the error limit for the horizontal codewords is reset to the minimum value relative to the number of ECC redundancy symbols in the horizontal codewords, and CHANGE_V is reset to FALSE at step 57. In the preferred embodiment, the horizontal codewords comprise 10 ECC redundancy symbols and the minimum error limit is 3. If at step 53 a correction was not made during the previous vertical pass (CHANGE_V FALSE) and at step 59 the error limits (V_LIM and H_LIM) have reached their maximums, then the product code is uncorrectable since no further corrections are possible and the system exits unsuccessfully. Otherwise, if at step 61 the error limit H_LIM for the horizontal codewords is less than its maximum (in this case 5), then it is incremented by one at step 63 to increase the correction power of the code for the next horizontal pass.

[0082] Continuing now with the flow diagram of FIG. 5A for executing the horizontal pass, at step 44 the P/Q decoder generates the ECC syndromes for the current horizontal codeword at ROW. If the ECC syndromes indicate that the codeword contains errors at step 87, the ECC syndromes are evaluated at step 46 to determine whether the codeword is correctable. As explained above with reference to FIG. 3B, the codeword is deemed uncorrectable if the UNC detector 23 determines that the number and location of the errors are uncorrectable, or if the NUM ERRS detector 31 detects a number of errors greater than the current error limit 35 for the horizontal codewords (H_LIM of FIG. 5E). If the codeword is not correctable, then an erasure pointer is generated and added to the current erasure polynomial σ(x)_EPat step 48 by executing the flow diagram of FIG. 5C, described below, and the HORIZ. ERRS variable is incremented at step 50. If the codeword is correctable at step 46, then at step 52 a branch is executed depending on VERT. ERRS, the number of uncorrectable column codewords encountered during the previous vertical pass. If VERT. ERRS is greater than 10, then the erasure polynomial σ(x)_EPHgenerated during the previous vertical pass cannot be used to correct the horizontal codewords. Instead, at step 54 an error locator polynomial σ(x) is generated using the ECC syndromes generated at step 44. If the VERT. ERRS is not greater than 10 at step 52, then the erasure polynomial σ(x)_EPHgenerated during the previous vertical pass can be used to correct the horizontal codeword, therefore the erasure polynomial σ(x)_EPHis assigned to the error locator polynomial at step 56. The current horizontal codeword is then corrected at step 58 using the error locator polynomial and the ECC syndromes generated at step 44 (see FIG. 3A and above discussion), and the variable CHANGE_H is set to TRUE to indicate that a correction was made during the current horizontal pass.

[0083] At step 60, the PVAL register is multiplied by α⁻¹, thereby decrementing the erasure pointer location l so that it corresponds to the next symbol in the vertical codewords. The ROW variable is incremented at step 62 to the next row codeword, and the above procedure is re-iterated until ROW equals 208 at step 64 indicating that all of the row codewords have been processed. If at step 66 the product code has been successfully corrected after the horizontal pass, then the process simply exits. Otherwise, the current erasure polynomial σ(x)_EPis saved at step 68 by assigning it to σ(x)_EPV(i.e., the contents of the coefficient registers σ₁-σ₁₆of FIG. 41, are transferred to a set of storage registers). The saved coefficients of the erasure polynomial σ(x)_EPVare then used to correct the column codewords during the next vertical pass concurrent with generating a new erasure polynomial σ(x)_EPusing the coefficient registers σ₁-σ₁₆and circuitry of FIG. 4A.

[0084] The flow diagram executed during a vertical pass is shown in FIG. 5B. Before each vertical pass, the error limit for the vertical codewords is adjusted at step 65 by executing the flow diagram of FIG. 5F. Referring to FIG. 5F, at step 70 a VERT. ERRS variable representing the number of uncorrectable column codewords encountered is initialized to zero, a COL variable is initialized to zero to point to the first column codeword, and the PVAL register is initialized to α¹⁸¹corresponding to erasure location l=181 in a horizontal codeword. If at step 71 the number of uncorrectable codewords encountered in the previous horizontal pass is not greater than 16, then the erasure polynomial is used to correct the vertical codewords and the error limit V_LIM for the vertical codewords is set to 16 at step 73. Otherwise, at step 75 a branch is executed depending on whether a correction was made during the previous horizontal pass. If a correction was made (CHANGE_H==TRUE), then at step 77 the error limit for the vertical codewords is reset to the minimum value relative to the number of ECC redundancy symbols in the vertical codewords, and CHANGE_H is reset to FALSE at step 85. In the preferred embodiment, the vertical codewords comprise 16 ECC redundancy symbols and the minimum error limit is 6. If at step At 75 a correction was not made during the previous horizontal pass (CHANGE_H==FALSE) and at step 79 the error limits (V_LIM and H_LIM) have reached their maximums, then the product code is uncorrectable since no further corrections are possible and the system exits unsuccessfully. Otherwise, if at step 81 the error limit V_LIM for the vertical codewords is less than its maximum (in this case 8), then it is incremented by one at step 83 to increase the correction power of the code for the next vertical pass.

[0085] Continuing now with the flow diagram of FIG. 5B for executing the vertical pass, at step 72 the P/Q decoder generates the ECC syndromes for the current column codeword at COL. If the ECC syndromes indicate that the codeword contains errors at step 89, the ECC syndromes are evaluated at step 74 to determine whether the codeword is correctable. As explained above with reference to FIG. 3B, the codeword is deemed uncorrectable if the UNC detector 23 determines that the number and location of the errors are uncorrectable, or if the NUM ERRS detector 31 detects a number of errors greater than the current error limit 35 for the vertical codewords (V_LIM of FIG. 5F). If the codeword is not correctable, then an erasure pointer is generated and added to the current erasure polynomial σ(x)_EPat step 76 by executing the flow diagram of FIG. 5D, described below, and the VERT. ERRS variable is incremented at. step 78. If the codeword is correctable at step 74, then at step 80 a branch is executed depending on HORIZ. ERRS, the number of uncorrectable row codewords encountered during the previous horizontal pass. If HORIZ. ERRS is greater than 16, then the erasure polynomial σ(x)_EPVgenerated during the previous horizontal pass cannot be used to correct the vertical codewords. Instead, at step 82 an error locator polynomial σ(x) is generated using the ECC( syndromes generated at step 72. If the HORIZ. ERRS is not greater than 16 at step 80, then the erasure polynomial σ(x)_EPVgenerated during the previous horizontal pass can be used to correct the vertical codeword, therefore it is assigned to the error locator polynomial at step 84. The current vertical codeword is then corrected at step 86 using the error locator polynomial and the ECC syndromes generated at step 72 (see FIG. 3A and above discussion), and the variable CHANGE_V is set to TRUE to indicate that a, correction was made during the current vertical pass.

[0086] At step 88, the PVAL register is multiplied by α⁻¹, thereby decrementing the erasure pointer location l so that it corresponds, to the next symbol in the horizontal codewords. The COL variable is then incremented at step 90 to the next column codeword, and the above procedure is re-iterated until COL equals 182 at step 92 indicating that all of the column codewords have been processed. If at step 94 the product code has been successfully corrected after the vertical pass, then the process simply exits. Otherwise, the current erasure polynomial σ(x)_EPis saved at step 96 by assigning it to σ(x)_EPH(i.e., the contents of the coefficient registers σ₁-σ₁₆of FIG. 4A are transferred to a set of storage registers). The saved coefficients of the erasure polynomial σ(x)_EPHare then used to correct the row codewords during the next horizontal pass concurrent. with generating a new erasure polynomial σ(x)_EPusing the coefficient registers σ₁-σ₁₆and circuitry of FIG. 4A.

[0087] The flow diagram executed to update the erasure polynomial σ(x)_EPduring a horizontal pass (step 48 of FIG. 5A) is shown in FIG. 5C. If at step 98 the number of uncorrectable horizontal codewords HORIZ. ERRS is greater than 15, then control simply returns to FIG. 5A because there are already too many uncorrectable horizontal codewords-the erasure polynomial σ(x)_EPcannot be used to correct. the column codewords during the next vertical pass if there are more than 16 erasure pointers. If HORIZ. ERRS is not greater than 15 at step 98, then at step 100 a counter variable i, which tracks the 16 iterations of the above-described “A phase” and “B phase”, is initialized to zero. At step 102 a counter A, which tracks the 8 clock cycles of the “A phase”, is initialized to zero. Then at step 104, bit A of the σ₁₆coefficient register 26₁₆of FIG. 4A is updated by adding its contents to the inner product of PVAL and the σ₁₅coefficient stored in the coefficient register 26₁₅as described above. The PVAL register is then updated by multiplying its contents by α at step 106, and the counter variable A is incremented at step 108. The above procedure is then re-iterated until A equals 8 at step 110 indicating the end of the “A phase”.

[0088] At the beginning of the “B phase”, a variable B which tracks the 8 clock cycles, is initialized to zero at step 112. Then at step 114, bit B of the σ_kcoefficient register 26_kis shifted into bit B of σ_k+1coefficient register 26_k+1. At step 116, the PVAL register is multiplied by α⁻¹, and at step 118, the counter variable B is incremented. The above procedure is re-iterated until B equals 8 at step 120 indicating the end of the “B phase”.

[0089] At step 122, the counter i is incremented and the A and B phases re-iterated until i equals 16 at step 124 indicating the end of the update procedure to add an erasure pointer for an uncorrectable row codeword. During the last iteration (i=15) as described above, at: step 104 the bits of the σ₁₆coefficient register 26₁₆are simply added to bit 7 of PVAL since the σ₀coefficient of the erasure polynomial σ_EP(x) is always equal to 1.

[0090] The flow diagram executed to update the erasure polynomial σ(x)_EPduring a vertical pass (step 76 of FIG. 5B) is shown in FIG. 5D. The steps are essentially identical to that of FIG. 5C, except, that at step 126 control returns to FIG. 5B if the number of uncorrectable vertical codewords VERT. ERRS exceeds 9, indicating that the erasure polynomial σ_EP(x) cannot be used to correct the row codewords during the next horizontal pass because there are only 10 ECC redundancy symbols in the row codewords (i.e., up to 10 row codeword symbols can be corrected). Also, the A and B phases are re-iterated only 10 times to update the 10 coefficients of the erasure polynomial σ_EP(x) generated during a vertical pass.

[0091] Although the erasure polynomial update procedures of FIGS. 5C and 5D require 16 and 10 iterations, respectively, of the A and B phases to add a single erasure pointer, typically they do not, introduce any additional latency because the circuitry of FIG. 4A operates concurrent with the error correction procedure of FIGS. 5A and 5B. Furthermore, the number of uncorrectable codewords encountered is typically small and cannot exceed the maximum for the vertical and horizontal passes (10 and 16) otherwise the update procedure is bypassed. Consequently, the complexity and cost of the circuitry in FIG. 4A is minimized by using bit-serial techniques without compromising the overall speed of the correction system.

[0092] In fact, during the last pass over the product code (horizontal or vertical) when the erasure polynomial σ_EP(x) is used to correct: the codewords, the correction speed actually increases because it is not necessary to generate the conventional error locator polynomial σ(x) at step 54 of FIG. 5A and at step 82 of FIG. 5B. Furthermore, it is not necessary to generate an erasure polynomial σ_EP(x) for each codeword; a single erasure polynomial σ_EPH(x) or σ_EPV(x) is used to correct all of the codewords (step 56 of FIG. 5A and step 84 of FIG. 5B)

[0093] Thus, the objects of the invention have been fully realized through the embodiments disclosed herein. Those skilled in the art will appreciate that the various aspects of the invention can be achieved through different embodiments without departing from the essential function. For example, the product code shown in FIG. 2 is typically employed in digital video discs (DVDs), but the present invention is equally applicable to other product code formats, including the format used in compact discs (CDs). Furthermore, the present invention could be applied to other multi-dimensional codes, not just product codes. Thus, the particular embodiments disclosed are illustrative and not meant to limit the scope of the invention as appropriately construed by the following claims.