CORHESPONDENCE 737

Solution: The weight distribution of Hamming (8, 4) code of cl,,, = 4 = 28, is l+‘(8) = W(0) = 1, W(4) = 14 where W(j) denotes the number of codewords of weight (j). Therefore, the 14 codewords of weight 4 are selected. It is interesting to note that this set is actually of maximum size.

Some other examples of obtaining calling station codes are those codes derived from Hadamard matrices [4], [lo], and codes derived from maximum-minimum distance codes [3], [5].

Let us consider first the case in which n 5 2(n - k) - b. Suppose the error is

E(X) = XjB(X) (1)

where B(X) is of degree 5 b - 1. We note that R(X) modulo G(X) is itself the error if the error is confined to the first n - k digits. This accounts for errors starting at X0, X’, X2, . . . , or Xv-k-b.

ACKNOWLEDGMENT

The problem was originally suggested by K. Y. Sih, and the author benefited by discussions with Dr. J. E. MacDonald who has independently found many constructive methods to obtain minimum separation codes [5]. The author is also very grateful to the reviewer for his valuable comments.

M.Y. HSIAO~ University of Florida Gainesville, Fla.

REFERENCES

On the other hand, if the error is confined to the last n - k digits, then R*(X) modulo G*(X) gives E*(X), where

R*(X) = Xv-‘R $ 0 (2)

G*(X) = X”-kG $ 0 (3)

E*(x) = +y’E $ . 0 (4)

[l] D. L. Cohn end J. M. German, “A code separation property,” IEEE Trans. Information Theory (Correspondence), vol. IT-S, pp. 382-383, October 1962.

[2] C. V. Freiman, “Protective block codes for asymmetric binary channels,” IBM Corp., Poughkeepsie, N. Y., Research Rept. RC-457, May 1961.

[3] J. E. MacDonald, “Design methods for maximum minimum-distance error- correction codes,” IBM J., pp. 43-57, January 1960.

[4] E&W. Peterson, Error Correctzng Code. Cambridge, Mass.: M.I.T. Press, [5] J. E.’ MacDonald, “Maximal code construction for communications channels

with symmetric or asymmetric noise,” Illinois, Urbana, October 1965.

Ph.D. dissertation, University of IS] J. MacWilliams, “A theorem on the distribution of weights in a systematic

code,” Bell Sys. Tech. J., vol. 42, p. 79, January 1963. 171 J. M. Go&h&, “Analysis of weight distribution in binary cyclic codes,”

IEEE Trans. Information Theory (Correspondence), vol. IT-12 pp. 401-402, July 1966.

This accounts for errors starting at Xk, Xk+r, or X*-l. If n s 2(n - k) - 5, then k 6 n - k - b, and all positions are ac- counted for.

[S] H. D. Goldman, M. Kliman, and H. Smola, “The weight structure of some BoseChaudhuri codes,” IEEE Trans. Information Theory (Correspondence). vol. IT-l& pp. 167-169, January 1968.

[Q] J. E. Levy,“4 weight distribution bound for linear codes,” IEEE Trans. Information Theory, vol. IT-14, pp. 187-190, May 1968.

IlO] S. W. Golomb, Ed., Digital Communications with Space Applications. wood Cliffs, N. J.: Prentice-Hall, 1964.

Engle-

This procedure can be generalized as follows. Let u = n - k - b + 1, and let T be the smallest integer such that TU $ n - k 2 n.

Step 0: Compute Ro(X) = R(X) modulo G(X). If RI(X) is correctable, it gives the error. If not, go to Step 1.

Step 1: Compute Ri(X) = XUR~-~(X) modulo G(X). If R,(X) is correctable, XuiRi(X) is the error pattern. Otherwise, do Step i + 1. If no correctable error pattern is found by Step r, the error is not a burst of length b or less and cannot be corrected.

Step 0 accounts for any burst that ends in the first n - k symbols. Step i, i > 0, accounts for any burst not previously found, and ending at or before the (n - k f iu)th symbol. With the above choice for r, any burst of length b or less will be found. The number of steps required is r f 1 at most.

3 Now with IBM Corp., Systems Development Division, Poughkeepsie, N. Y. 12602.

ACKNOWLEDGMENT

Decoding of Binary Cyclic Burst-Error-Correcting Codes

Let V be an (n, k) binary cyclic code capable of correcting all burst of length b or less [l]-[3]. The object of this corre- spondence is to give a procedure for decoding V that requires approximately k/(n - k - b) f 2 steps. This procedure is partly based on an already available technique.

Manuscript received June 17, 1968; revised April 30. 1969. This work wss supported in part by the National Research Council of Canada under Grant A-3371.

[2] P. Fire, “A class of multiple error correcting binary codes for non-independent errors,” Sylvania Reconnaissance Systems Laboratory, Mountain View, Calif. Sylvania Rept. R SL-E-2, 1959.

[3] W. W. Peterson, Error correcting codes. New York: Wiley, 1961, pp. 183-185. -on In, IOU-IJI.

The authors sincerely appreciate the helpful comments of Profs. J. K. Wolf and W. W. Peterson of the University of Hawaii.

S. G. S. SHIVA TONY ZEITOUN Dept. of Elec. Engrg. University of Ottawa Ottawa 2, Ont., Canada

REFERENCES [l] N. Abramson, “Error correcting codes from linear sequential networks,” Proc.

4th London Symp. on Information Theory, (Butterworths, London), 1961, pp. 26-38.