公开(公告)号：US07346834B2

公开(公告)日：2008-03-18

申请号：US11157082

申请日：2005-06-20

Applicant: Lih-Jyh Weng

Inventor： Lih-Jyh Weng

IPC: H03M13/00

CPC classification number: H03M13/152 ,  G11B20/0021 ,  H03M13/00

Abstract: A system that produces one or more non-repeating randomizer sequences of up to 2m−1 or more m-bit symbols includes a randomizer circuit that is set up in accordance with a polynomial with primitive elements of GF(2m) as coefficients. The system combines the randomizer sequence with all the symbols of ECC code words that are encoded using a BCH code over GF(2m) to produce a randomized code word. The particular primitive elements used and/or an initial state of one or more registers in the system specifies the particular sequence produced by the system. The initial state of each of the one or more registers is a selected one of the 2m−1 elements of GF(2m), and thus, 2m−1 different sequences may be produced by selecting a different initial state for a given one of the registers. If the coefficients are also selected from, for example, a set of “p” possible values, the system produces p*(2m−1) different sequences. The system may thus be used to encrypt the ECC code word by associating the code word with a particular selected initial state and/or coefficient. The coefficients may be selected to produce randomizer sequences that are predetermined minimum distances away from both the ECC code words.

Abstract translation: 产生一个或多个高达2×1或更多个m位符号的非重复随机化器序列的系统包括根据具有GF的原始元素的多项式建立的随机化器电路 （2 ）作为系数。 该系统将随机化器序列与使用GF（2 ）上的BCH码编码的ECC码字的所有符号组合以产生随机码字。 系统中使用的特定原始元素和/或一个或多个寄存器的初始状态指定了系统产生的特定序列。 一个或多个寄存器中的每一个的初始状态是GF（2 ）的2×2个元素中的一个，因此， 可以通过为给定的一个寄存器选择不同的初始状态来产生不同的序列。 如果还从例如一组“p”可能值中选择系数，则系统产生不同序列的p *（2-m-1）。 因此，该系统可以用于通过将码字与特定的选择的初始状态和/或系数相关联来加密ECC码字。 可以选择系数以产生远离两个ECC码字的预定最小距离的随机化序列。

公开(公告)号：US06779011B2

公开(公告)日：2004-08-17

申请号：US09796051

申请日：2001-02-28

Applicant: Lih-Jyh Weng ,  Dana Hall ,  Christine Imrich

Inventor： Lih-Jyh Weng ,  Dana Hall ,  Christine Imrich

IPC: G06F772

CPC classification number: G06F7/726

Abstract: A system determines the multiplicative inverse of A∈GF(22M) by representing A using a selected basis in which basis elements are squares of one another, and performing various operations that involve raising A to powers of 2 as cyclic rotations of A. The system also performs multiplication operations over GF(22M) or subfields thereof by calculating the coefficients of the product of two elements A and B that are represented using the selected basis as combinations of the coefficients of cyclically rotated versions of A and B. The system further utilizes a relatively small look-up table that contains the multiplicative inverses of selected elements of a subfield of GF(22M). The system may then cyclically rotate the multiplicative inverse values read from the table to produce the multiplicative inverses of the remaining elements of the subfield. Thereafter, as applicable, the system further manipulates the multiplicative inverse of the subfield element, to produce the multiplicative inverse of the desired element of GF(22M). Using the selected basis, elements of GF(22M) that are elements of the subfields have m lowest-order coefficients that are duplicates of the m highest order coefficients. Each element in the look-up table can thus be represented using only m bits, and the table can be entered with m bits.

Abstract translation: 系统通过使用选择的基础来确定A∈GF（2 <2M>）的乘法逆，其中基元是彼此的平方，并且执行涉及将A提高到2的幂的各种操作作为A的循环旋转 该系统还通过计算使用所选择的基础表示的两个元素A和B的乘积的系数作为A的循环旋转版本的系数的组合来执行GF（2 <2M>）或其子场的乘法运算，以及 该系统进一步利用一个相对较小的查找表，其中包含GF（2 <2M>）的子域的选定元素的乘法反转。 然后，系统可以循环地旋转从表读取的乘法反向值，以产生子场的剩余元素的乘法反转。 此后，在适用的情况下，系统进一步操纵子场元素的乘法逆，以产生GF（2 <2M>）的期望元素的乘法逆。 使用所选择的基础，作为子场的元素的GF（2 <2M>）的元素具有m个最高阶系数的m个最低阶系数。 因此，可以仅使用m位来表示查找表中的每个元素，并且可以以m位输入表。

公开(公告)号：US06772390B2

公开(公告)日：2004-08-03

申请号：US09727838

申请日：2000-11-30

Applicant: Lih-Jyh Weng ,  Dana Hall

Inventor： Lih-Jyh Weng ,  Dana Hall

IPC: H03M1315

CPC classification number: H03M13/2909 ,  H03M13/151 ,  H03M13/1515

Abstract: A method of determining error values including loading an error correction code (ECC) entity having rows representing data symbols, determining an error location for a first row, generating an error syndrome for the first row, determining an erasure constant array from the error location, determining an error location for each of the remaining rows, generating an error syndrome for each of the remaining rows and determining the error values for each of the rows from the corresponding error location and corresponding error syndrome and the constant.

Abstract translation: 一种确定误差值的方法，包括加载具有代表数据符号的行的纠错码（ECC）实体，确定第一行的错误位置，产生第一行的错误校正，从误差位置确定擦除常数阵列， 确定每个剩余行的错误位置，为每个剩余行生成错误校正，并从相应的错误位置和相应的错误综合征和常数确定每行的错误值。

公开(公告)号：US06651214B1

公开(公告)日：2003-11-18

申请号：US09479312

申请日：2000-01-06

Applicant: Lih-Jyh Weng ,  Dana E. Hall

Inventor： Lih-Jyh Weng ,  Dana E. Hall

IPC: G11C2900

CPC classification number: H03M13/159 ,  H03M13/1515 ,  H03M13/158

Abstract: A bidirectional code decoding method and apparatus is presented. It uses a class of Reed-Solomon codes capable of bidirectional decoding, more specifically, those for which a value of L for a Galois Field element &agr;L is chosen as −(R−1)/2 for odd values of R and 2(m−1)−R/2 for even values of R. When the symbols of such codes are received at a decoder in a reverse order (from that in which the symbols are normally received) during a reverse directional read, the decoder produces reverse directional syndromes S˜(−k) and converts the reverse directional syndromes S˜(−k) to syndromes S(k) by multiplying S˜(−k) by &agr;(n−1)k for k=L, L+1, . . . , L+R−1. Alternatively, the decoder adjusts error location values for errors occurring in reverse order code word symbols to correspond to error location values that correspond to an error locations that would be determined if the symbols were to be received in the order in which the symbols are normally received.

Abstract translation: 提出了一种双向代码解码方法和装置。 它使用能够进行双向解码的一类里德 - 所罗门码，更具体地说，对于伽罗瓦域元素α的值L被选为R的奇数值的（R-1）/ 2， 在反向定向读取期间，当这些代码的符号在解码器处以相反的顺序（从符号被正常接收的符号）接收到时，2 <（m-1）> -R / 解码器产生逆方向综合征S（k），并将反向方向综合征S（k）转换为综合征S（k），将S（k）乘以α<（n- 1）k> k = L，L + 1，...。 。 。 ，L + R-1。 或者，解码器调整在逆序码字符号中发生的错误的错误位置值，以对应于错误位置值，该错误位置值对应于将按照正常接收符号的顺序接收符号将被确定的错误位置 。

公开(公告)号：US06640319B1

公开(公告)日：2003-10-28

申请号：US09650968

申请日：2000-08-29

Applicant: Lih-Jyh Weng

Inventor： Lih-Jyh Weng

IPC: H03M1315

CPC classification number: H03M13/15 ,  H03M13/33 ,  H04L7/0008 ,  H04L7/048

Abstract: An R-stage error correction system constructed in accordance with a distance d Reed-Soloman code performs a modified encoding step to include in the encoding a predetermined non-zero “coset symbol” that results in the encoder producing a code word for recording that includes a coset leader of a distance d-1 code. During the modified encoding step, the coset symbol is included in the value produced by the Rthstage of the encoder during the encoding of a code word data symbol or a first redundancy symbol. The coset symbol is thereafter included in the encoding of the remaining redundancy symbols when the value produced by the last stage is fed back to the preceding stages. During decoding, the system decodes the code word and the included coset leader to generate associated error syndromes. In a last decoding step, the system removes the effects of the included coset leader from the syndromes, by combining a predetermined syndrome value with the syndrome value generated in the Rthstage . If there is no synchronization error and the code word is error-free, the result is a set of all zero syndromes. If there are code word errors but no synchronization error, the system produces a non-zero syndrome pattern that is associated with a correctable number of errors. Otherwise, if there is a synchronization error, the inclusion of the predetermined syndrome value produces a syndrome pattern that is associated with an uncorrectable number of errors.

Abstract translation: 根据距离d Reed-Soloman码构建的R阶段纠错系统执行修改的编码步骤，以在编码中包括导致编码器产生用于记录的代码字的预定的非零“陪集符号”，其包括 距离d-1代码的陪集领袖。 在修改编码步骤期间，在代码字数据符号或第一冗余符号的编码期间，陪集符号被包括在由编码器的第R级产生的值中。 当最后一级产生的值被反馈到前一级时，陪集符号随后被包括在剩余冗余符号的编码中。 在解码期间，系统解码代码字和所包含的陪集引导器以产生相关联的错误综合征。 在最后一个解码步骤中，通过将预定的校正子值与在第R阶段中产生的校正子值相组合，系统从校正子中除去所包含的陪集前导的效果。 如果没有同步错误，并且代码字是无错误的，则结果是一组全部零综合征。 如果存在代码字错误但没有同步错误，则系统产生与可校正错误数量相关联的非零校验码模式。 否则，如果存在同步错误，则包含预定校正子值产生与不可校正错误数量相关联的综合征模式。

公开(公告)号：US06236340B1

公开(公告)日：2001-05-22

申请号：US09225259

申请日：1999-01-04

Applicant: Ara Patapoutian ,  Lih-Jyh Weng

Inventor： Ara Patapoutian ,  Lih-Jyh Weng

IPC: H03M700

CPC classification number: H03M13/31

Abstract: A modulation encoder includes a base conversion circuit that converts a partitioned input data stream from a first base representation in accordance with the size of groups of bits in the partitioned stream into a second base representation. The base conversion circuit includes a circuit to produce intermediate values of the partitioned stream in the second base representation and a residual value logic circuit that performs modulo-arithmetic on intermediate values modulo the second base representation, and a one's complement logic network fed by the residual value logic to produce output code words. A modulation decoder includes a one's complement logic circuit fed by modulation code words to produces residual value words; and a base conversion circuit that converts residual value words from a first base representation into a second base representation to provide original user data.

Abstract translation: 调制编码器包括基本转换电路，其将根据分区流中的比特组的大小的第一基本表示的分割输入数据流转换为第二基本表示。 基本转换电路包括用于产生第二基本表示中的分割流的中间值的电路和对第二基本表示进行模数的中间值执行模运算的残余值逻辑电路，以及由剩余部分馈送的补码逻辑网络 产生输出代码字的值逻辑。 调制解码器包括由调制码字馈送的补码逻辑电路以产生剩余值字; 以及基本转换电路，其将剩余值字从第一基座表示转换成第二基本表示以提供原始用户数据。

公开(公告)号：US5978956A

公开(公告)日：1999-11-02

申请号：US984698

申请日：1997-12-03

Applicant: Lih-Jyh Weng ,  Ba-Zhong Shen

Inventor： Lih-Jyh Weng ,  Ba-Zhong Shen

IPC: H03M13/15 ,  H03M13/00

CPC classification number: H03M13/6505 ,  H03M13/1575

Abstract: An error correcting system transforms a degree-five error locator polynomial .sigma.(x) into the polynomial w(y)=y.sup.5 =b.sub.2 y.sup.2 +b.sub.1 y+b.sub.0, where b.sub.1 =0 or 1, and y=.sigma.(x), and determines the roots of .sigma.(x) based on the roots of w(y). The polynomial w(y) has (2.sup.M).sup.2 solutions over GF(2.sup.M), rather than (2.sup.M).sup.5 solutions, since for any solution with b.sub.2 =h.sub.2, b.sub.0 =h.sub.0 and b.sub.1 =1, there is no such solution with b.sub.2 =h.sub.2, b.sub.0 =h.sub.0 and b.sub.1 =0. Conversely, if there is such a solution with b.sub.1 =0 there are no such solutions with b.sub.1 =1. The system can thus use a table that has 2.sup.2M entries and is addressed by {b.sub.2, b.sub.0 }. The table produces roots y=r.sub.i, i=0, 1, 2, 3, 4, and the system then transforms the roots y=r.sub.i to the roots of .sigma.(x) by calculating x=.sigma..sup.-1 (y). To further reduce the overall table storage needs, the table may include in each entry four roots r.sub.i, i=0, 1, 2, 3, and the system then calculates the associated fifth root r.sub.4 by adding the stored roots. The size of the look-up table can be even further reduced by (i) segmenting the Galois Field (2.sup.M) into conjugate classes; (ii) determining which of the classes contain values of b.sub.0 that correspond to solutions of w(y) with five distinct roots; (iii) representing each of these classes, respectively, by a single value of b.sub.0 '=(b.sub.0).sup.2.spsp.k ; and (iv) including in the table for each class only those solutions that correspond to representative values of b.sub.0 '. The table then contains a relatively small number of sets of roots of each of the classes, with each set associated with a particular value of b.sub.2 '=b.sub.2.sup.2.spsp.k. The roots of w(y) are determined by finding the value of k that produces b.sub.0 ' and b.sub.2 ', entering the look-up table using {b.sub.0 ', b.sub.2 '}, raising the roots r.sub.i ' produced by the table to the power -2.sup.k to produce y=r.sub.i, and then transforming the result into the roots of .sigma.(x) by x=.sigma..sup.-1 (y).

公开(公告)号：US5948117A

公开(公告)日：1999-09-07

申请号：US786894

申请日：1997-01-23

Applicant: Lih-Jyh Weng ,  Ba-Zhong Shen ,  Shih Mo

Inventor： Lih-Jyh Weng ,  Ba-Zhong Shen ,  Shih Mo

IPC: H03M13/00 ,  H03M13/15

CPC classification number: H03M13/151

Abstract: An error correction system includes an encoder that uses a modified Reed-Solomon code to encode w-bit data symbols over GF(2.sup.w+i) and form a preliminary code with d-1 (w+i+1)-bit redundancy symbols. The preliminary code word is modified as necessary to set for each symbol a selected i bits to the same value as a corresponding i+1.sup.st bit. The preliminary code word also includes R pseudo redundancy symbols that are required for decoding the modified code word. The i+1 bits are then truncated from each of the code word symbols, to form a code word with w-bit symbols. The Galois Field GF(2.sup.w+i) is selected such that the elements of the field can be represented by (w+i+1)-bit symbols that are determined by a polynomial h(x) modulo an irreducible polynomial p(x), which isp(x)=x.sup.w+i +x.sup.w+i-1 + . . . +x.sup.1 +x.sup.0,with the polynomial h(x) representing a primitive element. The encoder uses the lower weight representations of the (w+i+1)-bit symbols and performs multiplication and raising the symbols to powers of 2.sup.i as combinations of cyclic shifts and permutations that are readily performed in hardware. A decoder decodes the code word as (w+i+1)-bit symbols to take advantage of the simplified multiplication and exponentiation operations.

Abstract translation: 纠错系统包括使用经修改的里德 - 所罗门码对GF（2w + i）上的w位数据符号进行编码并用d-1（w + i + 1）位冗余符号形成初步码的编码器。 根据需要修改初始码字，以将每个符号设置所选择的i位与相应的i + 1位相同的值。 初始码字还包括用于解码修改的码字所需的R伪冗余符号。 然后从每个码字符号中截断i + 1位，以形成具有w位符号的码字。 选择伽罗瓦域GF（2w + i），使得场的元素可以由通过不可约多项式p（x）的多项式h（x）确定的（w + i + 1）位符号来表示， ，其为p（x）= xw + i + xw + i-1 +。 。 。 + x1 + x0，多项式h（x）表示一个原始元素。 编码器使用（w + i + 1）位符号的较低权重表示，并且执行乘法和将符号提升为2i的幂作为在硬件中容易执行的循环移位和排列的组合。 解码器将码字解码为（w + i + 1）位符号，以利用简化的乘法运算和求幂运算。

公开(公告)号：US5901158A

公开(公告)日：1999-05-04

申请号：US837752

申请日：1997-04-22

Applicant: Lih-Jyh Weng ,  Ba-Zhong Shen ,  Shih Mo

Inventor： Lih-Jyh Weng ,  Ba-Zhong Shen ,  Shih Mo

IPC: G06F11/10 ,  G11B20/18 ,  H03M13/15 ,  H03M13/00

CPC classification number: G06F11/1008 ,  H03M13/151 ,  G11B20/1809

Abstract: The encoder/decoder system uses encoder hardware to encode data symbols and form a data code word. To decode, the system uses the same encoder hardware to determine a residue r(x), i.e. ##EQU1## where C.sub.r (x) is the retrieved code word and g(x) is the generator polynomial. If the residue is all zeros, the ECC code word is error-free and the system need not calculate the error syndrome. If the residue is non-zero, the encoder hardware is used, with various switches in different settings, to include certain multipliers in and exclude other multipliers from the further decoding operations of encoding the residue symbols to produce partial error syndromes that are the coefficients of the error syndrome polynomial.

Abstract translation: 编码器/解码器系统使用编码器硬件对数据符号进行编码并形成数据码字。 为了解码，系统使用相同的编码器硬件来确定残差r（x），即，其中Cr（x）是检索的码字，g（x）是生成多项式。 如果残差全为零，则ECC代码字是无错误的，并且系统不需要计算错误综合征。 如果残差不为零，则使用编码器硬件，其中不同设置中的各种交换机将特定的乘法器包含在其中并且排除其他乘法器以及对残差码元进行编码的另外的解码操作，以产生作为系数的部分误差综合征 误差多项式。

公开(公告)号：US5710782A

公开(公告)日：1998-01-20

申请号：US580351

申请日：1995-12-28

Applicant: Lih-Jyh Weng

Inventor： Lih-Jyh Weng

IPC: G11B20/18 ,  G06F7/72 ,  H03M13/00 ,  H03M13/15

CPC classification number: H03M13/151 ,  G06F7/724

Abstract: A system determines the error locations of four errors in GF(2.sup.2m) by transforming a degree-four error locator polynomial ultimately into two quadratic equations, finding the solutions of these equations, and from these solutions determining the roots of an error locator polynomial. The system first manipulates the error locator polynomial, which is of the form: .sigma.(x)=.sigma..sub.4 x.sup.4 +.sigma..sub.3 x.sup.3 .sigma..sub.2 x.sup.2 +.sigma..sub.1 x+.sigma..sub.0 �1! into the form: .theta.(y)=y.sup.4 +.theta..sub.2 y.sup.2 +.theta..sub.1 y+.theta..sub.0 �2! where the .theta..sub.i 's are combinations of the coefficients of the terms of the error locator polynomial. The system has thus produced an equation in which the coefficient of the y.sup.3 term is 0. The system then factors .theta.(y), to produce .theta.(y)=(y.sup.2 +t*y+u)*(y.sup.2 +v*y+w), �3! where "*" represents multiplication. It then determines the values of t, u, v and w by equating the coefficients of the two expressions for .theta.(y) and solving first for the variable t, which is equal to v, and then for the variables w and u. Once the values of the variables are determined, the system solves two quadratic equations, one for each of the factors of equation 3. Based on these solutions, the system determines the four error locations associated with the degree-four error locator polynomial.

Abstract translation: 系统通过将四度误差定位多项式最终转换成两个二次方程，找到这些方程的解，并从确定误差定位多项式的根的这些解决方案来确定GF（22m）中四个误差的误差位置。 系统首先将错误定位多项式操作为：sigma（x）= sigma 4x4 + sigma 3x3 sigma 2x2 + sigma 1x + sigma 0 [1]形成：theta（y）= y4 + theta 2y2 + theta 1y + theta 0 [ 2]其中θi是误差定位多项式的项的系数的组合。 因此，系统产生了一个方程，其中y3项的系数为0.然后，系统将θ（y）归因于θ（y）=（y2 + t * y + u）*（y2 + v * y + w），[3]其中“*”表示乘法。 然后，通过将θ（y）的两个表达式的系数相等于等于v的变量t，然后对于变量w和u求解，来确定t，u，v和w的值。 一旦确定变量的值，系统就可以求解两个二次方程，对于方程3的每一个因素，求解一个二次方程。根据这些解，系统确定与四度误差定位多项式相关联的四个误差位置。