US20010016052A1

US20010016052A1 - Method for encoding bits in a signal

Info

Publication number: US20010016052A1
Application number: US09/817,422
Authority: US
Inventors: Matthew Miller
Original assignee: NEC Laboratories America Inc
Current assignee: NEC Corp; NEC Laboratories America Inc
Priority date: 1998-06-26
Filing date: 2001-03-26
Publication date: 2001-08-23
Anticipated expiration: 2018-06-26
Also published as: US6438252B2; EP1108288A1; US6263087B1; WO2000001076A8; EP1108288A4; CA2336099A1; WO2000001076A1; JP2002519927A

Abstract

A message in the form of a sequence of bits is encoded into relatively short, non-binary signals. The message (or vector) is divided into two halves. The encoder first determines an encoding of the message into a vector half the length of the eventual encoded vector. The half-length vector forms the first half of the encoded, and is negated to form the second half of the encoded vector. An identical synchronization signal is added to both halves of the encoded vector. The synchronization signal is used to compensate for the possibility that the received vector (or message) is cyclically rotated from the original vector. During decoding, the decoder first obtains the synchronization signal by adding the two halves of the vector together, canceling the encoded message. The synchronization signal is used to find the original cyclical shift of the vector. The decoder obtains the half-length encoded message vector by taking the difference between the two halves of the aligned full-length encoded vector. After decoding the vector into a bit sequence, the encoder re-encodes that bit sequence and compares the resulting vector with the original vector to determine whether the original vector truly encoded a message or was simply a noise signal.

Description

FIELD OF THE INVENTION

The present invention relates to digital watermarking of data including image, video and multimedia data. Specifically, the invention relates to a method for encoding messages into relatively short, non-binary signals. It further relates to a method for decoding the messages from signals while simultaneously distinguishing signals that include encoded messages from signals that are simply noise. It is intended for application in storing information into media watermarks, but it also has application in other areas.

BACKGROUND OF THE INVENTION

U.S. patent application Ser. No. 08/534,894, filed Sep. 28, 1995, entitled “Secure Spread Spectrum Watermarking in Multimedia Data”, and U.S. patent application Ser. No. 08/746,022, filed Nov. 5, 1996, entitled “Digital Watermarking”, which are both incorporated herein by reference, describe methods of hiding and detecting relatively short, possibly non-binary vectors, or “watermarks”, in various kinds of media, such as images, audio sequences, and video clips. Such methods are also described in article by Cox et al. entitled “Secure Spread Spectrum Watermarking for Images, Audio and Video” in Proc. IEEE Int. Conf. On Image Proc, ICIP '96, Lausanne, 1996, vol. III, pp. 243-246. The detection methods described in those inventions and in the Cox et al. article are capable of testing for the presence or absence of only one such hidden vector or watermark at a time. In applications where a given portion of media contains more than one hidden vector, each encoding a different message, the detector must sequentially test for each possible vector and identify the one that is most likely. This is feasible for small numbers of possible messages, but becomes too costly when the number of messages becomes large.

If the data hiding methods referred to above are to be used for messages of more than a small number of bits, such as 4, then methods must be devised for a) encoding bit sequences into vectors suitable for data hiding, b) decoding vectors that are extracted from portions of media, and c) determining whether such extracted vectors represent true hidden data or are merely noise signals. These problems are analogous to those found in transmitting data across noisy channels, where the vectors that are hidden in images are analogous to the modulated signals sent across the channels. Because of this analogy, the terms “vector” and “signal” will be used interchangeably.

Standard methods of transmitting binary messages across possibly noisy channels involve two basic steps at the transmitter. First, the message is encoded with an error-correction code. This increases the number of bits in the message, introducing a pattern of redundancy that can be exploited at the receiver to correct errors resulting from noise in the channel. Second, the expanded message is modulated into a signal with physical characteristics suitable for transmission over the channel, by, for example, modulating the amplitude of a carrier signal (AM), modulating the frequency of a carrier signal (FM), or spread-spectrum coding. While standard error-correction codes are directly suitable for the coding of messages for the watermarking methods referred to above, the standard methods of modulation are not so well suited. Since they encode bits sequentially, they have difficulty encoding more than a small number of bits into the relatively short vectors used in electronic watermarks. For example, consider coding 32 bits into a watermark vector of length 64. Each bit would be coded with only two samples, which is not enough to reliably carry information using AM, FM, or spread-spectrum techniques.

An alternative method of converting the error-correction-encoded bits into a vector suitable for a watermark is to treat each bit as a separate channel in a multi-channel communications system. In such a system, each channel is assigned a separate carrier or spread-spectrum signal, and the modulated messages for all the channels are added together into a single signal. At the receiver, separate correlations are performed for each of the channels, and the correlations are compared against a threshold, to obtain each of the transmitted messages in parallel.

The multi-channel approach can be applied to watermarking by assigning a separate “reference signal” to each bit. These reference signals are designed to have low correlation with one another. The watermark is constructed by either adding or subtracting each reference signal according to whether the corresponding bit is 1 or 0, respectively. Alternatively, the watermark might be constructed by either adding or not adding each reference signal according to whether the corresponding bit is 1 or 0, respectively. Watermarks are decoded by computing their correlations with the reference signals, and comparing the correlations with a pair of thresholds to determine whether the corresponding bits are 1 or 0. Approaches of this type have been applied to watermarking in, for example, the following publications: Swanson, et al., entitled “Robust Data Hiding for Images” in Proc. IEEE Digital Signal Processing Workshop, Loen, Norway, Sept., 1996; Hartung, et al., entitled “Digital Watermarking of Raw and Compressed Video” in Digital Compression Technologies and Systems for Video Communications, Oct. 1996; and Smith, et al., entitled “Modulation and Information Hiding in Images” in Proc. Int. Workshop on Information Hiding, May 1996.

A principal problem with these approaches is that they are limited to only one type of detection region for the possible watermarks. In the prior art, the region of watermark space that is decoded to a given bit sequence results in detection regions that are all planes, intersecting at right angles. Such a rectangular detection region is appropriate for many data transmission systems, but for some systems, other shapes of detection regions are desirable. This is particularly true of watermarking.

The present invention solves the problem of hiding messages of 8 or more bits into media. The invention provides a method of encoding a message, described as a sequence of a given number, B, of bits, into a vector suitable for hiding in media by means of the data hiding methods referred to above. This encoding method is a procedure by which one of 2 ^Bpossible vectors is generated. The invention further provides a method of decoding a given, possibly distorted, vector into the sequence of B bits that is most likely encoded by the bit sequence. Finally, the invention also provides a method for quickly deciding whether a given vector is one of the 2^Bvectors that can be generated by the encoding process, or whether the vector is simply a noise signal. This last feature of the invention is important in deciding whether or not a given piece of media contains hidden information at all.

The detection regions (described below) employed in the detection of hidden data can be fairly arbitrary when practicing the present invention.

While the present invention solves the problem of hiding messages in media, it has application in other areas as well. In particular, the invention is suitable for use in any application where rectilinear detection regions are sub-optimal.

SUMMARY OF THE INVENTION

A primary object of the invention is a decoder which, after decoding a vector into a bit sequence, re-encodes that bit sequence and compares the resulting vector with the original vector to determine whether the original vector truly encoded a message or was simply a noise signal. A principal advantage of this approach is that it allows any type of comparison test to be performed, resulting in any shape of detection region. By proper design of the comparison test, an optimal detection region can be created.

A second object of the invention is a method of adding and using a synchronization signal to the encoded vector without disturbing the detectability of the encoded message. The synchronization signal is used to compensate for the possibility that the received vector is cyclically rotated from the original. The method splits the vector into two halves. During encoding, the encoder first determines an encoding of the message into a vector half the length of the eventual encoded vector. This half-length vector forms the first half of the encoded vector, and is negated to form the second half of the encoded vector. Then the encoder adds an identical synchronization signal to both halves of the encoded vector. During decoding, the decoder first obtains the synchronization signal by adding the two halves of the vector together, canceling the encoded message. The synchronization signal is used to find the original cyclical shift of the vector. Then the decoder obtains the half-length encoded message vector by taking the difference between the two halves of the aligned, full-length encoded vector.

Further objects of the invention will become more clearly apparent when the following description is read in conjunction with the accompanying drawing.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 shows detection regions for four possible two-bit messages that are created by prior-art methods of detecting signals. [0014]
FIG. 2 shows set of detection regions that are more desirable than those of FIG. 1, and that are possible when using the present invention. [0015]
FIG. 3 shows an alternative set of detection regions that are more desirable than those of FIG. 1, and that are possible when using the present invention. [0016]
FIG. 4 is a data flow diagram for a preferred method of encoding a message into a signal according to the present invention. [0017]
FIG. 5 is a data flow diagram for a preferred method of decoding a message according to the present invention. [0018]

DETAILED DESCRIPTION OF THE INVENTION

Referring now to the figures and to FIG. 1 in particular, there is shown detection regions created by separate thresholding of the correlations between separate, orthogonal reference signals and a received signal, as is the case in many prior-art systems. The figure shows a slice of high-dimensional space in which every vector corresponds to a possible received signal. The X- and Y-axes ([0019] 101,102) represent two orthogonal reference signals used to encode two bits. If it is assumed that the transmitter sends each bit using a predetermined amplitude of the reference signals, then it is possible to identify four possible combined signals that might be sent, depending on whether the message is 00, 01, 10, or 11. These four vectors are represented, respectively, by the four dots in the figure (103,104,105 and 106). The received vector is decoded by computing its correlation with each of the two reference signals. If the correlation with the low-order bit reference signal, i.e. the X axis 101, is above a threshold 107, then the low-order bit is 1. If it is below a threshold (108), then the low-order bit is 0. If it is between the thresholds 107 and 108, then the low-order bit cannot be determined and the received signal is assumed to be noise. Likewise, the correlation with the high-order bit reference signal, i.e. the Y-axis (102), is compared against two thresholds 109 and 110, to determine whether the high-order bit is 1, 0, or indeterminate. The resulting four detection regions are shown in gray (111,112,113 and 114).
The detection regions of FIG. 1 are suitable for systems that satisfy two criteria: a) the dimensionality of the space is significantly larger than the number of bits in the messages, and b) the distribution of noise vectors that might be added to a watermark vector is rotationally symmetric around that vector. In such a system, the noise vectors are expected to be orthogonal to all the reference signals, so they will not change the correlations between the resulting, received signals and the reference signals. However, in watermarking with relatively short vectors, neither of these criteria applies. As a result, detection regions of different shapes are more suitable. [0020]
Consider, first, systems in which the number of bits is close to or equal to the dimensionality of the space (i.e. the number of elements in the watermark vector). In such cases, noise vectors cannot be expected to have low correlations with all the reference signals. This means that a given noise vector will be likely to change the correlation between the received vector and at least one of the reference signals, and the change in correlation will often be enough to cross the threshold for the corresponding bit. A preferable set of detection regions for such a system is shown in FIG. 2. Here, there is a separate detection threshold ([0021] 201,202,203 and 204) for each of the four possible messages. A noise vector added to a given combined signal is likely to be orthogonal to that one combined signal, and, hence, unlikely to change the correlation between the received signal and the combined signal. Thus, it is unlikely to cross the threshold for that message.
Next, consider systems in which the distribution of noise vectors is not rotationally symmetric. The distribution of noise vectors applied to watermarks in images and other media can be highly asymmetric, and the best detection region to use for a single watermark is a hyperboloid. It follows that the best set of detection regions to use for a set of different possible watermarks is a set of different hyperboloids. Such detection regions are shown in FIG. 3. [0022]
Using the present invention, it is possible to obtain the detection regions shown in FIGS. 2 and 3, as well as other detection regions not specifically described here. [0023]
FIG. 4 is a data flow diagram of a preferred method of encoding a message into a signal according to the present invention. A message comprising a given number, B, of bits is provided to the [0024] encoder 401. The first step is to append a constant bit to the message 402. This bit always has a constant value, for example 1, and will be used during decoding to correct for errors that might flip all of the bits. The resulting, B+1 bit message is expanded by means of an error-correction encoder 403, which adds a number, P, of redundant parity bits. In the preferred embodiment, this expansion is performed by a Hamming code, but other codes, such as BCH, Reed-Solomon, or Convolutional codes, can also be used. The P+B+1 bits that are output from the error-correcting encoder are then sent to modulator 404. The modulator constructs a vector V′, of length 2L, by the following procedure:
1. Convert the P+B+1 bits into a column vector, D, of length P+B+1 in which each element is either a 1 or a −1 according to whether the corresponding bit is a 1 or a 0. [0025]
2. Multiply D by a “modulation matrix”, M, of size L rows by P+B+1 columns, to obtain a length L column vector, V. The exact design of M is generally arbitrary, except that its columns should have low correlation with one another. In the preferred implementation, the columns of M are chosen to have zero correlation with one another. [0026]
3. Create a [0027] length 2L column vector, V′, by appending the negation of V onto the end of V. That is $V^{'} = [\begin{matrix} + V \\ - V \end{matrix}]$
Finally, a [0028] synchronization signal 405, is added 406 to V′ to obtain the final, encoded vector 407. The synchronization signal 405 is a column vector, S′, of the form $S^{'} = [\begin{matrix} + S \\ + S \end{matrix}]$
where S is a column vector of length L that has good auto-correlation properties. [0029]
In alternative implementations, either one or both of the steps of appending a [0030] constant bit 402 and expanding the message by use of an error-correction encoder 403 may be eliminated. In such case, the length of the column vector, D, and the size of the modulation matrix, M, must be changed accordingly.
The output of the encoding process of FIG. 4 is an encoded vector that can be hidden in media using any of a variety of data hiding techniques. When the hidden data is to be detected, the data hiding technique first applies a procedure to extract a vector from the media. Then, rather than applying the test of the data hiding technique for the presence of a given watermark, the extracted vector is passed to the decoder of the present invention, as described below. [0031]
FIG. 5 is a data flow diagram of a preferred method of decoding vectors according to the present invention. A vector, R, is provided to [0032] decoder 501. This vector might be an output of the above-described encoder plus some noise resulting from a data-hiding process and/or other degradations, or the vector might be purely random. The function of the decoder is to determine whether R contains an encoded message, and, if so, to decode that message into a sequence of bits. In the figure, the first sequence of steps 503,504,505 and 506 is concerned with identifying the most likely message that might have been encoded in R. The remaining steps 507 and 508 are concerned with determining whether R actually contains that encoded message.
The first step in decoding the input vector is to align R with a [0033] synchronization signal 502. This is performed by the synchronizer 503. This step is required because the data hiding process that was used might result in extracted vectors that are cyclically shifted from their original, embedded form. Synchronization is performed by the following steps:
1. Assuming that R is an encoded message plus some noise, obtain a version of the half-length synchronization signal, S, by dividing R into two halves and averaging the halves together. Thus, we assume that [0034] $R = S^{'} + V^{'} + N = [\begin{matrix} S + V + N_{0} \\ S - V + N_{1} \end{matrix}]$
cyclically shifted by an unknown amount, where N=[N[0035] ₀ ^T,N₁ ^T]^Tis a random noise vector. The vectors S′ and V′ are as defined above. By splitting R into two halves and averaging both halves, we obtain
S _R=(S+V+N ₀ +S−V+N ₁)/2=(2S+N ₀ +N ₁)/2≅S
cyclically shifted by an unknown amount. [0036]
2. Find the cyclical rotation of S[0037] _Rthat has the highest correlation with S. Cyclically rotate R by the same amount to obtain the aligned version, R_A.
If R is an encoded message plus some noise, then the result of synchronization, R[0038] _A, is either aligned with the original, encoded message, or it is cyclically shifted by L. Thus, we have either $R_{A} = [\begin{matrix} S + V + N_{0} \\ S - V + N_{1} \end{matrix}] or$ $R_{A} = [\begin{matrix} S - V + N_{0} \\ S + V + N_{1} \end{matrix}]$
The next step is to demodulate the synchronized signal ([0039] 504) to obtain P+B+1 bits. Demodulation is performed by the following steps:
1. Obtain a version of the half-length modulated P+B+1 bits, V, by dividing R[0040] _Ainto two halves and averaging the first half with the negation of the second half. Thus, if R_Ais an encoded message, we have either
V _R=(S+V+N ₀ −S+V−N ₁)/2=(2V+N ₀ −N ₁)/2≅V
or [0041]
V _R=(S−V+N ₀ −S−V−N ₁)/2=(−2V+N ₀ −N ₁)/2≅−V
depending on whether R[0042] _Ais cyclically shifted by L.
2. Multiply V[0043] _Rby the inverse of M and threshold the result on 0 to obtain P+B+1 bits. Thus, wherever the result of this multiplication is greater than 0, we assign 1 to the corresponding bit. Wherever it is less than 0, we assign 0 to the corresponding bit.
If R[0044] _Ais an encoded message plus some noise, then the results of demodulation will be a reconstruction of the original P+B+1 bits, with some possible errors. The noise might cause some number of bits to be flipped. The possible cyclical shift by L might cause all the bits to be flipped.
The P+B+1 demodulated bits are then sent as input to the error-[0045] correction decoder 505. The decoder decodes the demodulated bits into B+1 bits. In the preferred embodiment, a Hamming decoder is used, but other error-correction decoders, such as BCH, Reed-Solomon or Convolutional decoders, might be used. However, if the encoder did not perform an error correction encoding step, then the error correction decoding step 505 is eliminated.
The final step in determining the most likely message encoded in R is to decide whether all the bits should be flipped to correct for the possibility that R[0046] _Ais cyclically shifted by L, which results in R_Ahaving the wrong sign. This step is performed by the sign corrector 506, which checks whether the constant bit added by the encoder was decoded as having the correct, constant value. For example, if the encoder always sets the constant bit to 1, but the bit has been decoded as 0, then all the bits must be flipped. The sign corrector 506 strips the constant bit after performing the correction. The output of the sign corrector 506 is the decoded message bits that are output from the decoder 510. However, if the encoder did append a constant bit to the message, then the sign correction step 506 is eliminated.
To decide whether R contains an encoded message, the output of [0047] sign corrector 506 is sent to re-encoder 507. Re-encoder 507 reproduces the encoding process of FIG. 4 in its entirety. The output of the re-encoder is the vector that would be used to encode the bits that are most likely encoded in R. This vector is then sent to comparator 508, which compares the vector with R using a test that results in the desired shape of detection region. In the preferred embodiment, the test is the hyperbolic test shown in FIG. 3. However, a wide variety of other tests are possible, including, but not limited to, thresholding on correlation (as shown in FIG. 2), on normalized correlation, or on correlation coefficient. The output of the comparator 509 is one bit indicating whether or not the decoded message 510 should be considered a true message or just the result of a noise signal.
While there has been described and illustrated preferred methods of encoding messages into and decoding messages from signals, it will be apparent to those skilled in the art that variations and modifications are possible without deviating from the broad teachings and spirit of the present invention which shall be limited solely by the scope of the claims appended hereto. [0048]

Claims

What is claimed is:

1. A method of encoding a message in the form of a sequence of bits to form an encoded vector comprising the steps of:

a) expanding the sequence of bits to obtain an expanded sequence of bits;

b) modulating the expanded sequence of bits to obtain a modulated vector;

c) appending the negation of the modulated vector to the modulated vector to obtain a double-length modulated vector, and

d) adding a synchronization signal to both the modulated vector and the negation of the modulated vector of the double-length modulated vector to obtain an encoded vector.

2. The method of

claim 1

, further comprising inserting the encoded vector into media as a watermark.

3. The method of

claim 1

, where said expanding comprises appending at least one redundant bit of constant value to the sequence of bits.

4. The method of

claim 3

, further comprising appending at least one additional redundant bit to the sequence of bits according to an error-correction encoding method.

5. The method of

claim 4

, further comprising inserting the encoded vector into media as a watermark.

6. The method of

claim 4

, where the error-correction encoding method is selected from the group consisting of Hamming encoding, BCH encoding, Reed-Solomon encoding, and Convolutional encoding.

7. The method of

claim 1

, where said expanding comprises appending at least one redundant bit to the sequence of bits according to an error-correction encoding method.

8. The method of

claim 7

, further comprising inserting the encoded vector into media as a watermark.

9. The method of

claim 7

, where the error-correction encoding method is selected from the group consisting of Hamming encoding, BCH encoding, Reed-Solomon encoding, or Convolutional encoding.

10. The method of

claim 1

, where said modulating comprises converting the expanded sequence of bits into a bit vector and multiplying the bit vector by a modulation matrix to obtain the modulated vector.

11. The method of

claim 10

, where the bit vector contains a first value corresponding to each 0 bit of the expanded sequence of bits and contains a second value corresponding to each 1 bit of the expanded sequence of bits.

12. The method of

claim 10

, where the columns of the modulation matrix have zero correlation with one another.

13. A method of decoding a received vector of two half vectors into a sequence of bits comprising the steps of:

a) averaging the first half of the received vector with the second half of the received vector to obtain a received synchronization vector;

b) determining an amount by which the received synchronization vector must be shifted to obtain the best match with a reference synchronization vector;

c) shifting the received vector by the determined amount to obtain an aligned received vector, and

d) demodulating the aligned received vector to obtain the sequence of bits.

14. The method of

claim 13

, where the received vector is a signal extracted from media according to a watermark extraction method.

15. The method of

claim 13

, where the determined amount is an amount of cyclical shifting.

16. The method of

claim 13

, where the match between the received synchronization vector and the reference synchronization vector is measured by computing the correlation between both synchronization vectors.

17. The method of

claim 13

, where said shifting the received vector is cyclical shifting.

18. The method of

claim 13

, where said demodulating comprises averaging a first half of the aligned received vector with the negation of a second half of the aligned received vector to obtain a received modulated vector; demodulating the received modulated vector to obtain an extended sequence of bits, including at least one redundant bit; and correcting errors in the extended sequence of bits using the at least one redundant bit in the extended sequence of bits to obtain the sequence of bits.

19. The method of

claim 18

20. The method of

claim 18

, where said demodulating the received modulated vector comprises multiplying the received modulated vector by the inverse of a modulation matrix to obtain a demodulated vector and comparing each element of the demodulated vector with a predetermined threshold to determine whether the corresponding bit in the sequence of bits should be a 0 or a 1.

21. The method of

claim 20

22. The method of

claim 20

, where the predetermined threshold is zero.

23. The method of

claim 18

, where said correcting errors is performed according to an error-correction decoding method.

24. The method of

claim 23

, where the error-correction decoding method is selected from the group consisting of Hamming decoding, BCH decoding, Reed-Solomon decoding, and Convolutional decoding.

25. The method of

claim 24

, where said correcting errors comprises comparing one or more redundant bits with constant values to determine whether all of the bits must be flipped.

26. The method of

claim 23

, where said correcting errors comprises comparing one or more redundant bits with constant values to determine whether all the bits must be flipped.

27. The method of

claim 18

28. A method of determining whether a received vector encodes a message in the form of a sequence of bits comprising the steps of:

a) decoding a received vector to obtain a received message;

b) encoding the received message to obtain an encoded vector, and

c) comparing the received vector with the encoded vector to determine whether the received vector encodes a message.

29. The method of

claim 28

30. The method of

claim 28

, where said decoding comprises averaging a first half of the received vector with a second half of the received vector to obtain a received synchronization vector; determining an amount by which the received synchronization vector must be shifted to obtain the best match with a reference synchronization vector; shifting the received vector by the determined amount to obtain an aligned received vector, and decoding the aligned received vector to obtain the sequence of bits.

31. The method of

claim 30

32. The method of

claim 30

, where the determined amount is an amount of cyclical shifting.

33. The method of

claim 30

34. The method of

claim 30

, where said shifting the received vector is cyclical shifting.

35. The method of

claim 30

, where said decoding comprises averaging a first half of the aligned received vector with the negation of a second half of the aligned received vector to obtain a received modulated vector; demodulating the received modulated vector to obtain an extended sequence of bits, including at least one redundant bit; correcting errors in the extended sequence of bits using the at least one redundant bit in the extended sequence of bits to obtain the sequence of bits.

36. The method of

claim 35

37. The method of

claim 35

, where said demodulating comprises multiplying the received modulated vector by the inverse of a modulation matrix to obtain a demodulated vector, and comparing each element of the demodulated vector with a predetermined threshold to determine whether the corresponding bit in the sequence of bits should be a 0 or a 1.

38. The method of

claim 37

39. The method of

claim 37

, where the predetermined threshold is zero.

40. The method of

claim 35

41. The method of

claim 40

, where the error-correction decoding method is selected from the group consisting of Hamming decoding, BCH decoding, Reed-Solomon decoding, or Convolutional decoding.

42. The method of

claim 41

, where said correcting errors comprises comparing the at least one redundant bit with a constant value to determine whether all of the bits must be flipped.

43. The method of

claim 42

44. The method of

claim 40

45. The method of

claim 35

, where said correcting errors comprises comparing the at least one redundant bit with constant a value to determine whether all of the bits must be flipped.

46. The method of

claim 45

47. The method of

claim 28

, where said encoding comprises appending at least one redundant bit to the received message to obtain an expanded sequence of bits; modulating the expanded sequence of bits to obtain a modulated vector; appending the negation of the modulated vector to the modulated vector to obtain a double-length modulated vector; and adding a synchronization vector to both the modulated vector and the negation of the modulated vector to obtain the encoded vector.

48. The method of

claim 47

49. The method of

claim 47

, where one or more redundant bits of constant value are appended to the expanded sequence of bits.

50. The method of

claim 49

, where further redundant bits are appended to the expanded sequence of bits according to an error-correction encoding method.

51. The method of

claim 50

52. The method of

claim 47

, further comprising appending additional redundant bits to the expanded sequence of bits according to an error-correction encoding method.

53. The method of

claim 52

54. The method of

claim 47

, where said modulating comprises converting the extended sequence of bits into a bit vector and multiplying the bit vector by a modulation matrix to obtain the modulated vector.

55. The method of

claim 54

, where the bit vector contains a first value corresponding to each 0 bit of the extended sequence of bits and contains a second value corresponding to each 1 bit of the extended sequence of bits.

56. The method of

claim 54

57. The method of

claim 28

, where said comparing comprises computing the correlation between the received vector and the encoded vector, and comparing the correlation with a predetermined threshold.

58. The method of

claim 28

, where said comparing comprises the steps of computing the correlation coefficient between the received vector and the encoded vector, and comparing the correlation coefficient with a predetermined threshold.

59. The method of

claim 58

60. The method of

claim 28

, where said comparing comprises the steps of computing the normalized correlation between the received vector and the encoded vector, and comparing the normalized correlation with a predetermined threshold.

61. The method of

claim 60

62. The method of

claim 28

, where said comparing determines whether the received vector is inside or outside a hyperboloid centered on the encoded vector.

63. The method of

claim 62

64. A method of encoding a message in the form of a sequence of bits to form an encoded vector comprising the steps of:

a) modulating the sequence of bits to obtain a modulated vector;

b) appending the negation of the modulated vector to the modulated vector to obtain a double-length modulated vector, and

c) adding a synchronization signal to both the modulated vector and the negation of the modulated vector of the double-length modulated vector to obtain an encoded vector.

65. The method of

claim 64

66. The method of

claim 64

, where said modulating comprises converting the sequence of bits into a bit vector and multiplying the bit vector by a modulation matrix to obtain the modulated vector.

67. The method of

claim 66

, where the bit vector contains a first value corresponding to each 0 bit of the expanded sequence of bitsand contains a second value corresponding to each 1 bit of the expanded sequence of bits.

68. The method of

claim 66