A decoding method for a (23,12) Golay code is extended to the important 1/2-rate (24,12) Golay code so that three errors can be corrected and four errors can be detected. It is shown that the method can be extended to any decoding method which can correct three errors in the (23,12) Golay code

Reed, I. S.

Truong, T.-K.

Yin, X.

English

NASA Technical Reports Server

TDA Progress Report 42-97
N89-27891
January-March 1989
Decoding of 1/2-Rate (24,12) Golay Codes
T.-K. Truong
Communications Systems Research Section
I. S. Reed and X. Yin
Department of Electrical Engineering, University of Southern California
In this article, a decoding method for a (23,12) Golay code is extended to the impor-
tant 1 2-rate (24,12) Golay code.
I. Introduction
The 23-bit Golay code is a very useful code, particularly for
those applications where a parity bit is added to yield a 1/2-
rate code. Recently a simplified procedure was developed for
decoding this important (24,12) Golay code [1]. It is shown
here that this procedure can be extended to any decoding
method which can correct three errors in the (23,12) Golay
code.
There are several known decoding procedures for correct-
ing the three possible errors of (23,12) Golay code:
(1) The minimum-distance method [2]
(2) The standard-array method [2], [3]
(3) The majority-logic method, suggested by Goethals [4]
(4) Kasami's error-trapping algorithm [5]
(5) The shift-and-search procedure [1]
(6) Algebraic algorithms which include Berlekamp's method
[2], [6] as well as the extended Bose-Chaudhuri-
Hocquenghem (BCH) algorithm suggested recently by
M. Ella in [7]
In this article, a simple method is given to extend any of these
procedures to the 1/2-rate (24,12) Golay code so that three
errors can be corrected and four errors can be detected.
II. Theorem for Close-Packed Error-
Correcting Codes
The (23,12) Golay code is a close-packed error-correcting
code for which the following theorem can be proved:
Theorem: Let C be the set of codewords of a Golay code.
Also let _. be the set of vectors of weight i. Then for any
c _ C, and anye 4 E E4, there is a £' E C such that
where£3 _ E 3.
c+e. = £'+£3
Proof: See Appendix and [1].
III. The (24,12) Golay Code
A (24,12) Golay codeword can be formed by adding an
even or odd parity-check bit to the (23,12) Golay codeword.
It is well known that such a (24,12) Golay code has the mini-
mum distance dmi n = 8. Thus any decoding algorithm can be
extended to the (24,12) Golay code with the correction of
three or fewer errors and the detection of four errors.
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There is no loss in generality to assume that the parity of a
(24,12) Golay codeword is even. That is, the sum of the 24
bits modulo 2 is equal to zero. Now assume during trans-
mission that four errors are added to the codeword. There are
two cases to consider:
(1) If the four errors occur in the first 23 bits, then by the
Theorem in Section lI, the addition of an error vector
of Hamming weight 4 to a codeword produces a 23-
bit vector that is equal to some other (23,12)Golay
codeword plus an error vector of weight 3. Thus if one
of the decoding algorithms in Section I is applied to
the first 23 bits, an error vector of weight 3 is added to
the received codeword. As a consequence the parity of
the (24,12) codeword also is changed. Hence by check-
ing the parity of the decoded codeword, the decoder
detects the presence of four errors.
(2) On the other hand, if three errors occur in the first 23
bits, and one error occurs in the parity bit, the decod-
ing algorithm corrects the three errors in the first 23
bits. The parity of the 23-bit decoded codeword now
differs from the received parity bit. Hence the decoder
detects the presence of four errors.
The extended decoding algorithm for the (24,12)Golay
code is summarized as follows:
Apply any decoding algorithm which can correct three
errors to the first 23 bits. If the number of errors is less
than three, the decoding procedure terminates normally.
If the number of errors is greater than or equal to three,
the parity of the decoded codeword is compared with
the received parity bit. If they are different, the decoder
detects four errors. The detailed flowchart of this decod-
ing procedure is shown in Fig. 1.
IV. An Example of Implementation
A complete algebraic decoding algorithm for the (23,12)
Golay code was found recently by M. Ella [7]. To illustrate
this method, define
E(x) = e22x22 +e21x21 +.--+elx +e 0
to be the error polynomial. Then the received codeword has
the form
R(x) = C(x) + E(x) = q(x)g(x) +g(x)
Suppose that three errors occur in the received code word
R(x) and assume that 2t _ d - 1. Since c_, a 3, and ¢x9 are the
roots of g(x), one has
s 1 = E(a) = R(a), s 3 = E(o_ 3) = R(o_3), s9 = E(a 9) = R(a 9)
where Sl, s3, and s 9 are called the syndromes of the code.
Assume zl, z2, and z 3 are the positions of the three errors.
Then the error-locator polynomial is defined by
3
o(z) = 1-I (z-zi) -- z3 +% z2 *%z+°3
i=1
where O 1 = gl + Z2 ÷ g3' 02 = 212"2 + Z2Z3 + 217"3' and 03 =
z lz 2z 3. Then from [7],
O 1 = S1
2 +DID
02 = S1
03 = $3 + Sl D1/3
where
D =
($3 + S3)2 + (S9 + $9 )
Hence under different conditions for the syndromes, the error
location polynomial has the following forms:
o(z) =
If S1 = S3 = S9 = 0
then no error
3 and s3 = S9Z + S1 If S3 = S1
then one error
Z2 + $1 Z
+ (s_ +D 1/3)
Z3 + $1 22
+(S_ +D'I3)z
+(s 3 + stD 1/3)
Ifs t :_s a and s3 = slD*/3
then two errors
Otherwise three errors
203
Note that the cube root Dl/3 is in GF(211) and can be com-
puted recursively, i.e.,
O1/3 = 01365 = O21°+28÷26+24+22+2°=
D 21°D28D26D24D22 D
After using the Chien search to find the error locations, one
can apply the decoding procedure in this article to decode the
(24,12) Golay code as described in Fig. 2.
A shift-and-search procedure for decoding Golay codes is
given in [1]. In a computer simulation, this procedure is corn-
pared with the Elia algebraic technique in terms of CPU time.
For three and four errors this comparison shows that the alge-
braic method is more than twice as fast as the shift-and-search
method in [1]. These results are shown in detail in Table 1.
V. Conclusion
The procedure given in this article extends the decoding of
the (23,12) Golay code to the (24,12)Golay code. This exten-
sion generalizes to any close-packed error-correcting code.
However, since there is only one other nontrivial multiple-
error-correcting perfect code-the (11,6)Golay code over
GF(3)-such a generality may be somewhat academic.
Acknowledgment
The authors thank Professor Van Lint, presently visiting at Caltech, for bringing
M. Elia's paper to our attention.
References
[1] T.-K. Truong, J. K. Holmes, 1. S. Reed, and X. Yin, "A Simplified Procedure for
Decoding the (23,12) and (24,12) Golay Codes," TDA Progress Report 42-96,
vol. October-December 1988, Jet Propulsion Laboratory, Pasadena, California, pp.
49-58, February 15, 1989.
[2] J. H. van Lint, Introduction to Coding Theory, New York: Springer-Verlag, 1982.
[3] R. McEliece, The Theory of Information and Coding, Reading, Massachusetts:
Addison-Wesley, 1977.
[4] J. J. MacWilliams and W. J. A. Sloane, The Theory of Error-Correcting Codes, Amster-
dam: North-Holland, 1977.
[5] S. Lin and D. J. Costello, Error Control Coding: Fundamentals and Applications,
Englewood Cliffs, New Jersey:Prentice-Hall, 1983.
[6] E. R. Berlekamp, Algebraic Coding Theory, New York: McGraw-Hill, 1968.
[7] M. Elia, "Algebraic Decoding of the (23,12) Golay Code," 1EEE Trans. Inform.
Theory, vol. IT-33, pp. 150-151, January 1987.
204
Table 1. Computer CPU time for decoder simulations
Number of errors
CPU time, ms 0 1 2 3 4
Algebraic method 15.5 16 20 21 22
Shift-and-search method 15 15.5 20 44.5 47
205
(23,t2) GOLAY CODE IPrl
T
START
1
ANY 123,12) GOLAY CODE
DECODING PROCEDURE
WHICH CORRECTS 3 ERRORS
c_ COMPUTE
PA RITY O F .¢..',p
]NO
END
Fig. 1. Flowchart for decoding (24,12) Golay codes.
I cr (z) : z + sI _=JNUMBER OF ERRORS = 1
START
1
YES
YES_
I COMPUTE D1/3 I
YES
ro
o'(z) = z3 + s1 z2 + (s 2 + D1/3)z I
+(s3+s 1 D 1/3)
NUMBER OF ERRORS=3
I ROOT SEARCH I
1
END
i NOERRO SPROG AMITERMINATES
Fig. 2. Flowchart of Elia's algebraic procedure.
206
Appendix
Proof of Theorem
It is well known that the (23,12) Golay code is a close-
packed code, i.e.. the following equation holds:
Let C be the set of codewords of the Golay code. Also let E i
be the set of vectors of weight i. Therefore, for any £ _ C, and
any e 4 E E4, then x = c + _e4 = C' + e where weight ofe satis-
fies w(e) _< 3. Hence e = x + c' so that by property w(x +y)
Iw(x) - w(,v)[ one has the inequality
w(e_) = w(_c c +e4)/>lw(£+c')-w(e4)l
where w(£ + _c') = dis(cc.,£'). Since minimum distance of the
code is dmi n = 7, one has finally that w(e__)/> 3, and the theor-
em is proved.
207
Appendix
Proof of Theorem
It is well known that the (23,12) Golay code is a close-
packed code. i.e., the following equation holds:
Let C be the set of codewords of the Golay code. Also let E i
be the set of vectors of weight i. Therefore, for any cE C, and
any _e4 E E4, then x = c + e 4 = C' d- e where weight ofe satis-
fies w(e) _< 3. Hence e = x + c'_ so that by property w(x +y)
Iw(x) - wO')l one has the inequality
d¢ rw(e) = w(g C__+e4)_lw_+c')-w(e4)l
where w(£ + £') = dis(c,£'). Since minimum distance of the
code is dmi n = 7, one has finally that w(e)/> 3, and the theor-
em is proved.
207


Decoding of 1/2-rate (24,12) Golay codes

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