The Art of Error Correcting Coding (Second Edition)
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But before checking machines could be seriously planned, the following problem — which is one, incidentally, of considerable interest from the standpoint of pure number theory — would require solution:. To construct, for a given finite field with elements, and for the checking therein of -element sequence , a reference matrix. Corresponding to any , and any — provided that is less than — there is a definite maximum.
This maximum should be ascertained, and the reference matrix therefore constructed. Backstory: Lester Saunders Hill wrote unpublished notes, about 40 pages long, probably in the mid- to late s. The notes were typewritten, but mathematical symbols, tables, insertions, and some footnotes were often handwritten.
Comments by transcriber will look like this: [This is a comment. I used Sage www. Here is just the 15th section of his paper. I hope to post more later. Part 14 is here. This matrix contains determinants of order with — determinants of first order being single elements of the matrix. We recall that if , is an element operand sequence in , a -element check based upon the matrix is:.
Let be a positive integer less than or equal to. If, in the transmittal of the sequence. In this statement, denotes the total number of elements in the field. The presence of error is evidently certain to be disclosed. There is no loss of generality of we assume [T his assumption implies.
The errors cannot escape disclosure. For, to do so, they would have to satisfy the system of homogeneous linear equations:.
Case 3: Let errors fall among the and errors fall among the. Without loss in generality, we may assume that the errors are , , affecting , and , , affecting ,. Writing , where denotes a non-negative integer which may or may not be zero, we have.
Hence of the are transmitted without error. But the matrix of coefficients in this system contains no vanishing determinant, so that the only solution would be. Assuming that the presence of error is to escape detection, we see, therefore, that the errors must satisfy linear equations as follows:.
The system determines all errors affecting the — that is, all the errors — as polynomial functions of the ,. Hence, all told, errors are determined as rational functions of the remaining errors. Hence the chance that the errors will check up, and thus escape disclosure, is only -in-. Here is just the 14th section of his paper. Part 13 is here. Illustration of checking in the field. We wish to provide checks upon arbitrary sequences , , …, of elements of any given finite algebraic field.
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Operations in the field can be used to illustrate all the essential points arising in this connection. Example 1 : Check of Type A upon a sequence , , …,. Suppose that we wish, in a certain message, to transmit the numerical sequence. The origin and significance of this sequence in the following form, in which it appears as a sequence of ten elements of :. A five-element check is obtained by the simple tabulation which follows. We use Table 4, noting that in each of the nine rows, the one-hundred columns are shown in four sections, there being twenty-five columns in each section.
For the first element, 38, in the operand sequence , place ruler under the proper section of row 1 of the Table. In row 1, read: 13 in column 38, 39 in column 13, 16 in column 39, 48 in column 16, 43 in column Start a tabulation:. For the second element, 46, in the operand sequence, place ruler under the proper section of row 2 of the Table. In row 2, read: 83 in column 46, 29 in column 83, 15 in column 29, 60 in column 15, 38 in column Continue the tabulation:.
Since the third and fourth elements of the operand sequence are equal to zero, skip them entirely. For the fifth element, 60, read in row 5 of the Table: 76 in column 60, 2 in column 76, 16 in column 2, 27 in column 16, 14 in column Proceed in this manner to the ninth element, 2, inclusive of the operand sequence.
There being no tenth row in the Table, simply add the tenth element, 99, of the operand sequence in each vertical column of the tabulation.
The art of error correcting coding - Robert H. Morelos-Zaragoza - Google книги
The full tabulation is:. The five-element check is therefore:. We transmit telegraphically the sequence of fifteen elements of :. A discussion of possible errors will be given in another section. If only a one-element check had been desired, we should have taken , and a one-column tabulation would have sufficed.
If a two-element check had been desired, we should have taken , and a two-column tabulation would have sufficed. If the operand sequence contains less than elements, the procedure, to determine check of Type A, is obvious. The manner of its evaluation, by means of Table 4, is clear. Example 2 : Check of Type B upon a sequence , , … ,. Suppose that we desire a five-element check upon the numerical sequence — that is, upon. We wish to evaluate. The tabulation is as follows:. Questions of rigor will be discussed in a subsequent section. It may be noted here, however, that our checks make very nice discriminations.
Thus, although the operand sequence in Examples 3 and 4 are closely similar, the checking sequences are widely different.
Art Error Correcting Coding by Morelos Zaragoza Robert
It will now be apparent to those who are acquainted with the problems of code communication that we are in a position to furnish powerful and economical checks on message sequences in any conceivable telegraphic system. We have only to partition messages into groups of not more than ten, or not more that eight, elements each — according to any definite prearrangement — and to check each group with a -check, as may be arranged , of Type A or of Type B.
If it is desired to work with longer groups, we have only to enlarge Table 4. The limitations introduced in this paper may be largely overcome, or removed, by suitable amplification of the Table employed. Yet Another Mathblog. Check out these references: R. Connelly and A. This is the information age with more and more telephone lines needed every day.
Want to reach out and touch someone? You need representation theory. This is cheating a little since the works in the reference below really use the theory of Lie groups instead of representation theory itself. Still, there is a tangential relation at least between representation theory of Lie groups and the solution to certain nonlinear network problems. Desoer, R. Brockett, J. Wood, R. Hirshorn, A. Willsky, G. Hollywood, CA, Control theory. Chirikjian and I. Antenna design. Hassibi, B. Shokrollahi, W. Design of stereo systems. Here are a few selected references. Blake and R.
Tillich and G.
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