Patent ID: 6970112

Claim:
A method for transmitting of a digital message which includes information elements of an additive Abelian group, comprising the steps of: coding the message, modulating the encoded message into a signal; transmitting the signal using a communication channel; demodulating the received signal into the encoded message; and decoding the encoded message into the message, wherein, in the coding step, the message is encoded according to a formula: Y n =X k {circle around (x)}G, where X k is a vector-row of the message which includes a k-number of the elements, Y n is a vector-row of the encoded message which includes the k-number of the elements and a check element, G is a generating matrix of operations which consists of a k-number of rows and a (k+1)-number of columns, the generating matrix G including one of a matrix Gâ€² and a matrix Gâ€³, where Gâ€² is generated by (a) a kÃ—k size matrix with operations g 0 diagonal positions and operations g 1 at other positions and (b) the (k+1)th column which is added to the kÃ—k size matrix from the right, the (k+1)th column representing a sequence of one of the operations g 0 and operations g 2 , and Gâ€³ is generated from Gâ€² by rearranging at least one of the rows and the columns, {circle around (x)} is an operation of a generalized matrix multiplication in accordance with one of the following rules: (i) y j = âŠ• Î£g v ij (x i ) for jâ‰¦k and y j =g 2 [ âŠ• Î£g v ij (x i )] for j=k+1, when the (k+1)th column corresponds to the sequence of operations g 0 , and (ii) y j = âŠ• Î£g v ij (x i ), when the (k+1)th column corresponds to the sequence of operations g 2 , where y j is a j-th element of the vector-row of the encoded message, âŠ• Î£g v ij (x i )=g v 1j (x 1 )âŠ•g v 2j (x 2 )âŠ• . . . âŠ•g v kj (x k ), âŠ• is an operation of summing the elements of the Abelian group, g v ij (x i ) is an operation g v for an element x i in accordance with a rule for a ij-th matrix element, v=[0,2], i=[1,k], j=[1,k+1], g 0 =x i âŠ•e, g 1 =x i âŠ•(âˆ’x i ), g 2 =x i âŠ•(âˆ’x i )âŠ•(âˆ’x i ), and e is a unity element of the Abelian group, wherein, in the decoding step, the encoded message is decoded by excluding from the vector-row Yâ€² n a particular element, a number of the particular element corresponding to the (k+1)th column, provided that H{circle around (x)}Yâ€² n T =e, where Yâ€² n T is a transposed vector-row Yâ€² n , and H is a check matrix of operations having a 1Ã—(k+1) size which is generated, using a transposition process of the (k+1)th column, by one of the following: (i) adding to the right of the (k+1)th column the operation g 0 and rearranging the columns, when the (k+1)th column includes the sequence of operations g 0 , and (ii) adding to the right of the (k+1)th column the operation g 2 and rearranging the columns, when the (k+1)th column includes the sequence of operations g 2 .