referring now to the figures and to fig1 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 ( 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 fig1 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 . 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 fig2 . here , there is a separate detection threshold ( 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 fig3 . using the present invention , it is possible to obtain the detection regions shown in fig2 and 3 , as well as other detection regions not specifically described here . fig4 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 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 . 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 . 3 . create a length 2l column vector , v ′, by appending the negation of v onto the end of v . that is v ′ = [ + v - v ] finally , a 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 ′ = [ + s + s ] where is a column vector of length l that has good auto - correlation properties . in alternative implementations , either one or both of the steps of appending a 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 fig4 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 . fig5 is a data flow diagram of a preferred method of decoding vectors according to the present invention . a vector , r , is provided to 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 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 r = s ′ + v ′ + n = [ s + v - n 0 s - v + n 1 ] cyclically shifted by an unknown amount , where n =[ n 0 t , n 1 t ] t is 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 { tilde over (=)} s 2 . find the cyclical rotation of sr that 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 a , is either aligned with the original , encoded message , or it is cyclically shifted by l . thus , we have either r a = [ s + v - n 0 s - v + n 1 ] or r a = [ s - v + n 0 s + v + n 1 ] the next step is to demodulate the synchronized signal ( 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 a into two halves and averaging the first half with the negation of the second half . thus , if r a is an encoded message , we have either v r =( s + v + n 0 − s + v − n 1 )/ 2 =( 2v + n 0 − n 1 )/ 2 { tilde over (=)} v v r =( s − v + n 0 − s − v − n 1 )/ 2 =(− 2v + n 0 − n 1 )/ 2 { tilde over (=)}− v depending on whether r a is cyclically shifted by l . 2 . multiply v r by 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 a is 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 - 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 a is cyclically shifted by l , which results in r a having 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 not 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 sign corrector 506 is sent to re - encoder 507 . re - encoder 507 reproduces the encoding process of fig4 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 fig3 . however , a wide variety of other tests are possible , including , but not limited to , thresholding on correlation ( as shown in fig2 ), 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 .