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1178 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001
TABLE IPARAMETERS [[n; k; d]] OF QUANTUM CODES FORm = 2; 3; 4.
S IS THE NONZERO SET OF THE RS CODE OVER GF(4 ).n = m(4 � 1); k = n� 2mjSj; d = jSj + 1
Corollary 5: Let C be a cyclic code over GF(4) of lengthnj(4m � 1) with zero setZ � f0; 1; 2; . . . ; n� 1g and nonzero setS = f0; 1; 2; . . . ; n � 1g n Z. If �2S mod n � Z, C is self-or-thogonal.
Example: Letn = 15. By Corollary 5, the(15; 6) cyclic code (overGF(4)) with zero set
Z = f0; 5; 10; 1; 4; 11; 14; 3; 12g
(zeros in GF(16)) will be self-orthogonal since the nonzeros
S = f7; 13; 2; 8; 6; 9g
are such that�2S mod n � Z. SinceS has four consecutive integers,the minimum distance of the dual of this code isd = 5. So, by Theorem1, a[[15; 3; 5]] quantum code may be obtained from this self-orthog-onal cyclic code.
A detailed table of quantum BCH codes can be found in [3].
B. Quantum Codes from 4-ary Images
The following corollary (to Theorems 3 and 4) may be used forconstruction of cyclic codes (over GF(4m)) whose GF(4)-images areHermitian self-orthogonal. LetZn = f0; 1; 2; . . . ; n � 1g. DivideZn into the cyclotomic cosets modulon under multiplication by4. LetCs be the coset represented bys.
Corollary 6: Let S be the nonzero set of a cyclic code of lengthnj(4m � 1) over GF(4m). ForS � Zn, let S = Zn n S andSc =
s2SCs. If �2Sc mod n � Sc, all images ofC are Hermitian self-
orthogonal.Proof: Sc and Sc are exactly the nonzero and zero sets of
Tm(C). By Corollary 5 and Theorem 3, the result follows.
As an example, consider the 16-ary(15; 4) cyclic code with nonzerosetS = f6; 7; 8; 9g. Hence,
Sc = f6; 9; 7; 13; 2; 8g
and
Sc = f0; 1; 4; 3; 12; 5; 10; 11; 14g:
Since
�2Scmodn = f3; 12; 1; 4; 11; 14g � Sc
all images ofC are self-orthogonal. Using Theorem 1, a[[30; 14; 5]]quantum code may be obtained from any GF(4)-image of this code.This code meets the lower bound specified in [1] for codeword lengthn = 30 and message lengthk = 14.
In general, a(n; k; d) code of lengthnj(4m�1) over GF(4m)withself-orthogonal images over GF(4) results in an[[mn; mn�2mk; d]]quantum code.
Table I is a partial list of the quantum codes that can be con-structed starting with Reed–Solomon (RS) codes over GF(4m) form = 2; 3; 4.
ACKNOWLEDGMENT
The authors wish to thank Dr. E. M. Rains, AT&T Research, for hishelp with interpretations in [1] and the two reviewers for providinghelpful comments.
REFERENCES
[1] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,“Quantum error correction via codes over GF(4),”IEEE Trans. Inform.Theory, vol. 44, pp. 1369–1387, July 1998.
[2] M. Grassl, W. Geiselmann, and T. Beth, “Quantum reed-solomoncodes,” inProc. AAECC Conf., 1999.
[3] M. Grassl and T. Beth, “Quantum BCH codes,” inProc. Int. Symp.Theoritical Electrical Engineering, Magdeburg, Germany, 1999, pp.207–212. LANL e-print quant-ph/9910060.
[4] K. Sakakibara, K. Tokiwa, and M. Kasahara, “Notes on q-ary expandedReed-Solomon codes over GF(q ),” Electron. Commun. in Japan, pt.3, vol. 72, no. 2, pp. 14–23, 1989.
[5] K. Sakakibara and M. Kasahara, “On the minimum distance of aq-aryimage of aq -ary cyclic code,”IEEE Trans. Inform. Theory, vol. 42,pp. 1631–1635, Sept. 1996.
[6] G. E. Séguin, “Theq-ary image of aq -ary cyclic code,”IEEE Trans.Inform. Theory, vol. 41, pp. 387–399, Mar. 1995.
[7] F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North-Holland, 1977.
New Binary Codes from a Chain of Cyclic Codes
Markus Grassl, Member, IEEE
Abstract—Starting with a chain of cyclic linear binary codes of length127, linear binary codes of lengths 129–167, and dimensions 30–50 are con-structed. Some of these codes have a minimum distance exceeding the lowerbound given in Brouwer’s table.
Index Terms—Binary linear codes, lower bounds, minimum Hammingdistance.
I. INTRODUCTION
In this correspondence, parameters of some new linear binary codesobtained by ConstructionX (see, e.g., [1] and below) are presented.Some of the codes improve the lower bound on the minimum distance
Manuscript received May 23, 2000; revised August 7, 2000.The author is with the Institut für Algorithmen und Kognitive Systeme
(IAKS), Universität Karlsruhe, 76128 Karlsruhe, Germany (e-mail: [email protected]).
Communicated by R. M. Roth, Associate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(01)01407-9.
0018–9448/01$10.00 © 2001 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001 1179
TABLE IIRREDUCIBLE FACTORS OFx � 1 OVER GF(2) WRITTEN AS MINIMAL
POLYNOMIALS � (x) OF� WHERE� IS A ROOT OFx + x + 1
of a linear binary codeC = [n; k] of lengthn and dimensionk ac-cording to Brouwer’s table [2]. In the following, this original lowerbound will be denoted bydBrouwer(n; k).
The main prerequisite to applying ConstructionX is to find a codeC1 and a subcodeC2 C1 with high minimum distance. Instead ofsearching for such pairs of codes individually, the following approachis taken. First, a chain of cyclic codes, i.e., a sequence of codesCi =[n; ki; di] of identical lengthn whereCi Ci+1, is computed. Then,taking appropriate subcodes, this chain is refined such that the refinedchain contains a code for each dimension0 � k � n.
The next section demonstrates how chains of cyclic codes can bederived exploiting the algebraic properties of cyclic codes. Then theparameters of the new codes are presented.
II. A CHAIN OF CYCLIC CODES OFLENGTH 127
Let � be a primitive element of the field GF(128) with minimalpolynomial�1(x) = x7 + x+1. The factors ofx127 � 1 over GF(2)are the minimal polynomials�i(x) of �i given in Table I. This fac-torization gives rise to a sequence of cyclic linear binary codesCj =[127; 127� 7j] where for0 � j � 18, Cj is generated by
gj(x) :=
j
i=1
fi(x): (1)
By construction,Cj+1 Cj . The parameters of the codesCj are listedin Table II. Together with the minimum distances, the Bose–Chaud-huri–Hocquenghem (BCH) boundsdBCH have been computed usingMAGMA [3]. While a lot of codes exceed the BCH bound, most of thecodes meet the lower bounddBrouwer. For j = 5; 6; 7, the minimumdistances of the codesCj are one less thandBrouwer, and the minimumdistance ofC8 is two less thandBrouwer.
By taking appropriate subcodes, the chain of cyclic codesCj can berefined. So there exists a chain of binary codesck = [127; k; d(ck)],1 � k � 127 with ck ck+1. If Cj+1 ck � Cj , then the minimumdistance of the codeck is at leastd(ck) = d(Cj).
TABLE IIPARAMETERS OF THECYCLIC CODESC = [127; k; d] GENERATED BY
g (x) (SEE (1))
TABLE IIIPARAMETERS OF THECOMPONENT CODESC = [127; k ; d ],
C = [127; k ; d ], AND C = [n ; k ; d ] USED TO OBTAIN THE
CODEC = [n; k; d] BY CONSTRUCTIONX
III. N EW CODES
The new codes are obtained by ConstructionX (see, e.g., [1, Ch. 18,Sec. 7, Theorem 9]), which is recalled for completeness.
Theorem 1 (ConstructionX): LetC2 = [n; k � k3; d2] be a sub-code of the codeC1 = [n; k; d1] and letC3 = [n3; k3; d3] be a thirdcode over the same alphabet. Then there exists a codeC = [n0; k; d0]with n0 = n + n3 andd0 � min(d2; d1 + d3).
1180 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001
TABLE IVUPDATED LOWER BOUNDS ON THEMINIMUM DISTANCES OFLINEAR BINARY CODESC = [n; k] FOR129 � n � 167 AND 30 � k � 50
For the codesC1 and C2 in ConstructionX, codesck and c`with c` ck from Section II are chosen. For the third codeC =
[n3; k3; d3]; an optimal code is taken, i.e., a code with maximalminimum distance for given length and dimension.
For 131 � n � 164 and30 � k � 50, several codes whose min-imum distance exceedsdBrouwer(n; k) were found. The parameters ofthose codes and of the component codes are listed in Table III.
In Table IV, the resulting updated lower bounds on the minimumdistances of linear binary codesC = [n; k] of lengthsn and dimen-sionsk in the ranges129 � n � 167 and30 � k � 50 are listed.
For unmarked entries, the original lower bound of Brouwer’s table [2]is repeated. Entries marked withX are obtained by ConstructionX.Entries marked withE are not directly obtained by ConstructionX,but by adding a parity-check symbol (and zeros) to a shorter code ofsame dimension. Similarly, entries marked withP andS are obtainedby puncturing (resp., shortening).
IV. DISCUSSION
From the factorization of the polynomialxn � 1 it is fairly easy toconstruct chains of cyclic codes of lengthn. It seems worthwhile to
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH 2001 1181
investigate for which lengths chains of codes can be constructed fromwhich codes with high minimum distance can be derived. On the otherhand, an explicit search for code triples fulfilling the conditions of Con-structionX may result in further improvements of the lower bounds onthe minimum distance.
ACKNOWLEDGMENT
The author wishes to thank T. Beth and D. Lazic for introducinghim to the theory of error-correcting codes. He also acknowledges thesupport by J. Cannon and A. Steel, members of the MAGMA group.
REFERENCES
[1] F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North-Holland, 1977.
[2] A. E. Brouwer, “Bounds on the size of linear codes,” inHandbook ofCoding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, TheNetherlands: Elsevier, 1998, vol. 1, pp. 295–461.
[3] W. Bosma, J. J. Cannon, and C. Playoust, “The Magma algebra systemI: The user language,”J. Symb. Comp., vol. 24, no. 3–4, pp. 235–266,1997.
Decoding the Quadratic Residue Code
Ruhua He, Member, IEEE, Irving S. Reed, Life Fellow, IEEE,Trieu-Kien Truong, Fellow, IEEE, and
Xuemin Chen, Senior Member, IEEE
Abstract—The techniques needed to decode the(47 24 11) quadraticresidue (QR) code differ from the schemes developed for cyclic codes in [1],[5], and [6]. By finding certain nonlinear relations between the known andunknown syndromes for this special code, two methods are developed to de-code up to the true minimum distance of the(47 24 11) QR code. Thesealgorithms can be utilized to decode effectively the -rate (48 24 12)QR code for correcting five errors and detecting six errors.
Index Terms—Conjugate class, cyclotomic coset, Newton identities, syn-drome.
I. INTRODUCTION
Let (n; (n+ 1)=2; d) denote a binary quadratic residue (QR) codewith generator polynomialg(x). The length of this code is a primenumber of the formn = 8m� 1, wherem is some integer. Let the
Manuscript received September 10, 1999; revised May 14, 2000. This workwas supported by the National Science Foundation under Grant NCR-9314347.
R. He is with Hughes Network System, San Diego, CA 92121 USA (e-mail:[email protected]).
I. S. Reed is with the Department of Electrical Engineering-Systems, EEB500, University of Southern California, Los Angeles, CA 90089-2565 USA(e-mail: [email protected]).
T.-K. Truong is with the College of Electrical and Information Engineering,I-Shou University, Kaohsiung County, Taiwan, 84008, R. O. C. (e-mail: [email protected]).
X. Chen is with Broadcom Corporation, Irvine, CA 92619 USA (e-mail:schen@ broadcom.com).
Communicated by R. Roth, Associate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(01)01523-1.
splitting field ofxn � 1 beGF (2k) for some integerk. The generatorpolynomialg(x) is given by
g(z) =i2Q
(z � �i)
where� 2 GF (2k) is a primitiventh root of unity andQ denotes theset of quadratic residues given by
Q = fiji � j2 mod n; j = 1; 2; . . . ; n� 1g:
Since2 2 Q, the setQ is closed under multiplication by two. Thus,Q can be represented by a disjoint union of cyclotomic cosets, modulon. These cyclotomic cosets are given by
Qr = fr2j mod njj = 0; 1; . . . ; nr � 1g
wherenr is the smallest positive integer such thatr2n � r modn,andr is the smallest element inQr. Then, sincen is a prime number ofthe form8m� 1, it is easy to see thatnrj(n� 1)=2. Elementr in Qr
is called the representative element of the cyclotomic cosetQr. Nextlet setS consist of all representatives of the QR code.S is called thebase set. Therefore
Q =r2S
Qr:
Now, let a codewordc(x) = a(x)g(x) be transmitted through anoisy channel to obtain a received word of the formr(x) = c(x) +e(x), wheree(x) is the polynomial of the received error pattern. ThesyndromesSi are defined by
Si = e(�i); for i � 0: (1)
If i 2 Qr, i.e.,i � r2n for some integerni, then, one has
Si = S2
r : (2)
Two syndromes with indexes in the same cyclotomic coset are said tobe conjugate to each other. The set of all syndromes that are conjugateto each other are called a conjugate class.
It is easy to see thatc(�i) = 0 for i 2 Q. Thus
Si = e(�i) = r(�i); i 2 Q:
Therefore, theSi’s for i 2 Q are the known syndromes. For the re-maining values ofi, theSi’s are unknown. One approach to the de-coding of a cyclic code is to find values of the unknown syndromesfrom the known syndromes [5], [6]. Lete(x) = �
j=1xr . Then, the
syndromesSi can be expressed in the form
Si = Zi1 + Zi
2 + � � �+ Zi� ; i � 0 (3)
whereZj = �r correspond with the error locationsrj , v � t fort = (dmin � 1)=2 anddmin is the minimum weight of the code. It iswell known that the mapping between the syndromesSi of a QR codeand the error patternse(x) of weight � t is one to one.
Next, define the error-locator polynomial of the error patterne(x) tobe
L(z) =
�
i=1
(z � Zi) = z� +
�
j=1
�jz��j (4)
0018–9448/01$10.00 © 2001 IEEE
