F.J. MacWilliams and N.J.A. Sloane. The Theory of Error Correcting Codes. North-Holland Publishing Company., 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....6 6 6 6 ( 1) 0 ( 0) 0 ( 0) 0 ( 0) 1 : 2 . n m 1) 7 7 7 7 7 7 7 7 5 By definition, this matrix has the property that any submatrix formed by deleting m rows of this matrix is invertible [17]. Moreover, any matrix derived from this matrix by a sequence of elementary matrix transformations maintains this property (since elementary matrix operations do not change the rank of a matrix [11] Therefore, constructing the matrix B is a simple matter of performing elementary transformations ....

F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes, Part I. North-Holland Publishing Company, Amsterdam, New York, Oxford, 1977.

....k # errors if given any codeword (v 1 , v n ) and any tuple (u 1 , u n ) over F such that 0 k # one can detect that (u 1 , u n ) is not a codeword. If the code is Maximal Distance Separable, then the maximum value of errors that can be detected is n 1 [11]. We say that the (k 1) outof n secret sharing scheme can correct k # errors if from any (v 1 , v n ) S(m) and any tuple (u 1 , u n ) over F with k # one can recover the secret m. If the code is Maximal Distance Separable, then the maximum value of errors that ....

....k # errors if from any (v 1 , v n ) S(m) and any tuple (u 1 , u n ) over F with k # one can recover the secret m. If the code is Maximal Distance Separable, then the maximum value of errors that allows the recovery of the vector (v 1 , v n ) is k 1) 2 [11]. A (k 1) out of n Maximal Distance Separable (MDS) secret sharing scheme is a (k 1) out of n secret sharing scheme with the property that for any k # 1) 2, one can correct k # errors and simultaneously detect n k k # 1 errors (as follows easily by generalizing [11, p. 10] Maximal ....

F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Company, 1978.

....to have a corrupted copy of a database held by host #. When the corruptions are non destructive, meaning that the corruptions only change data rather than adding new data or deleting old data, the problem of synchronizing the two databases is precisely the classical problem of error correction [22]. Many sources [23 27] have addressed synchronization of such non destructively corrupted databases. A more recent work [28] makes a direct link to coding theory by using a well known class of good codes known as Reed Solomon codes to affect such synchronizations. However, the applications that ....

F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, New York, 1977.

....to have a corrupted copy of a database held by host A. When the corruptions are non destructive, meaning that the corruptions only change data rather than adding new data or deleting old data, the problem of synchronizing the two databases is precisely the classical problem of error correction [22]. Many sources [23 27] have addressed synchronization of such non destructively corrupted databases. A more recent work [28] makes a direct link to coding theory by using a well known class of good codes known as Reed Solomon codes to affect such synchronizations. However, the applications that ....

F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, New York, 1977.

....computationally on the size of a codeword, which is exponential in b. The approach described above can nonetheless be used to design e#cient set reconciliation algorithms. Some error correcting codes, including Bose ChaudhuriHochquenghem (BCH) codes and their subclass of Reed Solomon (RS) codes [37], depend only weakly on the overall code size. For these types of codes, the computational complexity of encoding a message and decoding to an error locator polynomial depends only on the number of 1 s in a codeword and not on the codeword s length. The dependence on the length appears only in ....

....at evaluation 66 points p i to get values (p 3 ) # (p m ) 3.7.1) for an upper bound m on the number of tolerated di#erences with other sets. These evaluations are transmitted to a reconciling host. On the other hand, the redundant residue code formulation of Reed Solomon codes [37] involves converting a message u = u 1 , u 2 , u 3 , u k ) into a polynomial u(x) i=1 u i x i 1 . In this formulation, the codeword corresponding to the message u is given by c = u(# ) u(# ) where # is a primitive root of unity in F q . The Chinese ....

F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Company, New York, 1977.

....complexity. They presented an algorithm that, with high probability, corrected most cases of up to disagreeing pages using a single message of signatures, where the size of each signature was logarithmic in the overall le size n. Abdel Gha ar 13 and Abbadi [26] used Reed Solomon codes [29] to provide a deterministic algorithm for the same problem that required at most 2 signatures. In the direction of spurious error correction, Schwarz, Bowdidge, and Burkhard [25] extended Metzner and Kapturowski s result to the case of extraneous or missing pages. Using a divide andconquer ....

F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, New York, 1977.

....stands in contrast to the rich body of existing work that examines faults arising from noise and packet loss. This line of research has produced results ranging from fundamental limits on encoding in the presence of noise [39] to error detecting codes [18] forward and backward error correction [31], to sophisticated time out and retry algorithms [6, 24] However, all the studies fail to address the effect of channel faults on entities outside of the communication systems, e.g. the operating system and applications. Along the same lines, little is understood about the effect of ....

F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam, 1977.

....is odd . 2) 3 The proof will be given in the next subsection. How to construct such F will also be illustrated in the proof. Remark. Our design method is only for quadratic (n; m; k) SAC functions. if and n is even (3) or and n is odd : 4) Proof. From Gilbert Varshamov bound [12], there exists a linear [N; m; d] code if N Gammam d Gamma2 N Gamma 1 Then from Theorem 10, we obtain this bound. ut such that and m = n Gamma u Gamma 1. and m = n Gamma u Gamma 2. Proof. 1) Apply Theorem 10 to the Hamming code which is 8 x x 2 x 1 x 1 x 2 3 x 8 ....

F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Company, 1977.

....given any codeword (v 1 ; vn ) and any tuple (u 1 ; un ) over F such that 0 jfi : u i 6= v i ; 1 i ngj k one can detect that (u 1 ; un ) is not a codeword. If the code is Maximal Distance Separable, then the maximum value of errors that can be detected is n k 1 [12]. We say that the (k 1) out of n secret sharing scheme can correct k errors if from any S(M) v 1 ; vn ) and any tuple (u 1 ; un ) over F with jfi : u i 6= v i ; 1 i ngj k one can recover the secret m. If the code is Maximal Distance Separable, then the maximum ....

....from any S(M) v 1 ; vn ) and any tuple (u 1 ; un ) over F with jfi : u i 6= v i ; 1 i ngj k one can recover the secret m. If the code is Maximal Distance Separable, then the maximum value of errors that allows the recovery of the vector (v 1 ; vn ) is (n k 1) 2 [12]. A (k 1) out of n Maximal Distance Separable (MDS) secret sharing scheme is a (k 1) out of n secret sharing scheme with the property that for any k (n k 1) 2, one can correct k errors and simultaneously detect n k k 1 errors (as follows easily by generalizing [12, p. 10] Maximal ....

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F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Company, 1978.

.... short, are greedily constructed codes which were introduced by Levenshtein [9] and again by Conway and Sloane in [4, 5] Lexicodes have surprisingly good encoding parameters and include, among other famous optimal codes, the Hamming codes, the binary Golay code, and certain quadratic residue codes [5, 10]. Several authors [3, 5, 9] have proved that lexicodes are linear, and comparison with optimal linear codes of the same length and dimension shows that lexicodes are usually within one of the optimal minimum distance [5] and often exhibit the smallest known covering radius [6] Hence, lexicodes ....

F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Company, New York, 1977.

....remaining to the left of the boundary line, is O(lgn) The i 0 reasoning is as follows: IrmJ y hm(i) i=0 LrmJ LrmJ = y bm(i) q y Am(i) y bm(i) q [rmJ 1 = c2 [r cllgnJ ca lg n for appropriate constants c2 and c3. In order to determine 1, we make use of the following bound [22]: LJ where 0 1 2 and H(r) is the binary entropy function. 1 We now have: 1H(v) rlgr (1 r)lg(1 r) y bm(i) n 2 TM n 2mH(r) 2 TM n2m(H(r) l) lg n 1 This last quantity is O(1) when m 1 n(r) Thus if we pick Cl 1 n(r) then after m Cl lg n steps, there ....

F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes, volume 1, page 310. North Holland Publishing Company, 1977.

....N represent the block size of an SPN consisting of R rounds of n n s boxes (M per round) a simple example of an SPN with N = 16; n = 4; M = N = 4; and R = 3 is illustrated in Figure 1. An interesting class of linear transformations is the one based on Maximum Distance Separable (MDS) codes [7]. The use of such linear transformations was first proposed by Vaudenay in [13] and then utilized in the cipher SHARK [12] and later in the cipher SQUARE [2] This class of linear transformations has the advantage that the number of s boxes involved in any 2 rounds of a linear approximation or in ....

....can be used to perform both the encryption and the decryption operations [16] Rijmen et al. [12] noted that the framework of linear codes over GF (2 ) provides an elegant way to construct the linear transformation layer. More details about the theory of error correcting codes can be found in [7]. Let C be a (2M; M; d) code over GF (2 ) Let G = IjA] be the generator matrix in echelon form where A is a nonsingular M M matrix and I is the M M identity matrix. Then A defines an invertible linear mapping M : X Y = AX: 1) If the matrix A is used in the implementation of ....

[Article contains additional citation context not shown here]

F.J. MacWilliams and N.J.A. Sloane. The theory of error correcting codes. North-Holland Publishing Company, 1977.

....of chips remaining to the left of the boundary line, is O(lg n) The reasoning is as follows: hm (i) Delta m (i) b m (i) brmc 1 = c 2 br Delta c 1 lg nc c 3 lg n for appropriate constants c 2 and c 3 . In order to determine c 1 , we make use of the following bound [22]: bkc kH( where 0 1=2 and H(r) is the binary entropy function. We now have: mH(r) n2 m(H(r) Gamma1) This last quantity is O(1) when m = 1 GammaH(r) Thus if we pick c 1 = 1 GammaH(r) then after m = c 1 lg n steps, there will be at most c 3 lg n chips remaining ....

F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes, volume 1, page 310. North Holland Publishing Company, 1977.

....could be made robust using this generalized Reed Solomon code. However, when the number of errors is rather large, no algorithm is known to locate the errors in this generalized Reed Solomon code. Note that it is easy to prove that if one could efficiently generalize the BerlekampMassey algorithm [53] to work for this generalized Reed Solomon code, that the discrete log problem and factoring problem would both be easy [28] Whether there exists a polynomial time algorithm to detect the error locations is still an open problem. However, the problem need not to be addressed to obtain robust ....

F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Company, 1978.

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F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, NorthHolland Publishing Company, Amsterdam, 1978.

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F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, NorthHolland Publishing Company, Amsterdam, 1978.

....2t 1, there corresponds a one way data reconciliation and verification function for Hamming errors with transmission size (q 1) i. i 0 A consequence of Corollary 5 is that perfect codes produce optimal one way data reconciliation schemes. For example, the length 23 binary Golay code [20] can be used to produce an optimal scheme for reconciling subsets of Z3. 5 Examples and Applications 5.1 Groups under composition Our first examples are of error sets that form a group under composition, in which case several of the bounds in the paper meet. Example 2: Cyclic shift) Consider ....

F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North- Holland Publishing Company, New York, 1977.

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F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam, 1977.

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F.J. MacWilliams and N.J.A. Sloane. The Theory of Error Correcting Codes. North-Holland Publishing Company., 1977.

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F.J. MacWilliams and N.J.A. Sloane. The Theory of Error Correcting Codes. North-Holland Publishing Company., 1977.

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F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1978.

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F. J. MacWilliams, N. J. A. Sloane, The Theory of Error- Correcting Codes, North-Holland Publishing Company, 1977.

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F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company 1977.

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MacWilliams, F.J., Sloane, N.J.A., The Theory of Error-Correcting Codes, Part I, North-Holland Publishing Company, 1977.

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F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. NorthHolland Publishing Company, 1977.

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