IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007 1193

and q is even, and b(q � 1)m=2c = (q � 1)m=2 otherwise. The firstfew values are given by the following table:

m S(m; (q � 1)m=2) S(m; ((q � 1)m� 1)=2)

1 1 1

2 q -3 (3q2 + 1)=4 3q2=4

4 (2q3 + q)=3 -5 (115q4 + 50q2 + 27)=192 (115q4 + 20q2)=192

6 (11q5 + 5q3 + 4q)=20 -

We can now prove Theorem 3. By definition

A(q; r; !) = (� � 1)� + (q � 1)�

=(S(!; b(q � 1)!=2c)� 1)qr�! + (q � 1)(r� !):

In particular

A(q; r; 1) = (q � 1)(r� 1)

A(q; r; 2) = (q � 1)qr�2 + (q � 1)(r� 2)

A(q; r; 3) � (3q2 � 3)qr�3=4+ (q � 1)(r� 3):

Hence

A(q; r; 2)�A(q; r; 1) = (q � 1)qr�2 � (q � 1) � 0

for r � 2. Similarly

A(q; r; 2)�A(q; r; 3) � ((q2 � q)� (3q2 � 3)=4)qr�3 + (q � 1)

= (q � 1)(q� 3)qr�3=4+ (q � 1) > 0

for r � 3;

A(q; r; 3)�A(q; r; 4)

� (q(3q2=4� 1)� ((2q3 + q)=3� 1))qr�4 + (q � 1)

= (q3 � 16q + 12)qr�4=12+ (q � 1) > 0

for r � 4;

A(q; r; 4)� A(q; r; 5)

� (q � 1)(13q3 + 13q2 + 27q � 165)qr�5=192+ (q � 1) > 0

for r � 5;

A(q; r; 5)�A(q; r; 6)

� 47q5 � 140q3 � 1152q+ 960)qr�6=960 + (q � 1) > 0

for r � 6.

REFERENCES

[1] A. Ahlswede and H. Aydinian, “Error control codes for parallel asym-metric channels,” in Proc. IEEE Int. Symp. Information Theory, Seattle,WA, Jul. 2006, pp. 1768–1772.

[2] B. Bose, S. Elmougy, and L. G. Tallini, Systematic t-UnidirectionalError-Detecting Codes in Z unpublished manuscript, 2005.

[3] B. Bose and D. J. Lin, “Systematic unidirectional error-detectingcodes,” IEEE Trans. Comp., vol. C–34, pp. 1026–1032, 1985.

[4] B. Bose and D. K. Pradhan, “Optimal unidirectional error detecting/correcting codes,” IEEE Trans. Comp., vol. 31, no. 11, pp. 564–568,Nov. 1982.

[5] S. Elmougy, “Some Contributions to Asymmetric Control Codes,”Ph.D. dissertation, Oregon State Univ., Corvallis, 2005.

[6] T. Kløve, P. Oprisan, and B. Bose, “The probability of undetectederror for a class of asymmetric error detecting codes,” IEEE Trans. Inf.Theory, vol. 51, no. 3, pp. 1202–1205, Mar. 2005.

[7] P. A. MacMahon, Combinatory Analysis. New York: Chelsea, 1960,vol. I & II.

New Quasi-Cyclic Codes From Simplex Codes

Eric Z. Chen

Abstract—As a generalization of cyclic codes, quasi-cyclic (QC) codescontain many good linear codes. But quasi-cyclic codes studied so far aremainly limited to one generator (1-generator) QC codes. In this correspon-dence, 2-generator and 3-generator QC codes are studied, and many good,new QC codes are constructed from simplex codes. Some new binary QCcodes or related codes, that improve the bounds on maximum minimumdistance for binary linear codes are constructed. They are 5-generator QC[93; 17; 34] and [254;23;102] codes, and related [96; 17;36], [256;23;104]codes.

Index Terms—Binary linear codes, quasi-cyclic codes, simplex codes.

I. INTRODUCTION

A code is said to be quasi-cyclic if every cyclic shift of a codeword byp positions results in another codeword. Therefore quasi-cyclic (QC)codes are a generalization of cyclic codes with p = 1. It has been shownthat QC codes contain many good linear codes [1]–[4]. Unfortunately,there are not many construction methods for good QC codes. Lots ofresearchers have turned to the power of modern computers, and manygood QC codes which improve lower bounds on the minimum distanceof linear codes have been found [5]–[7]. The author maintains a data-base of best-known binary QC codes [8].

In [12], two 2-generator QC codes were presented. Very little re-search on how to construct good g-generator QC codes has been made,with g > 1. In this correspondence, construction methods are presentedto construct good 2-, and 3-generator QC codes from cyclic simplexcodes. Many good, new QC codes are found. And several codes thatimprove the bounds on maximum minimum distance for binary linearcodes are also presented.

II. CYCLIC CODES AND QC CODES

A. Cyclic Hamming Codes and Simplex Codes

A q-ary linear [n; k; d] code is a k-dimensional subspace of an n-di-mensional vector space over GF(q), with minimum distance d betweenany two codewords. A code is said to be cyclic if every cyclic shift of acodeword is also a codeword. A cyclic code is described by the polyno-mial algebra. A cyclic [n; k; d] code has a unique generator polynomialg(x). It is a polynomial with degree of n�k. All codewords of a cycliccode are multiples of g(x) modulo xn � 1.

It is well known that for any positive integer k, there is a binarycyclic simplex [n; k; d] code with the minimum distance d = 2k�1,where n = 2k � 1. It should be noted that binary simplex codes are

Manuscript received September 1, 2006; revised November 23, 2006.The author is with the School of Engineering, Kristianstad University, 291 88

Kristianstad, Sweden (e-mail: eric.chen@tec.hkr.se).Communicated by T. Etzion, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2006.890727

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1194 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007

equidistance codes where 2k � 1 nonzero codewords have a weight of2k�1.

B. Quasi-Cyclic Codes

A code is said to be quasi-cyclic (QC) if a cyclic shift of any code-word by p positions is still a codeword. Thus a cyclic code is a QCcode with p = 1. The block length n of a QC code is a multiple of p,or n = m � p.

Circulants, or cyclic matrices, are basic components in the generatormatrix for a QC code. Anm�m cyclic or circulant matrix is defined as

C =

c0 c1 � � � cm�1

cm�1 c0 � � � cm�2

cm�2 cm�1 � � � cm�3...

... � � �...

c1 c2 � � � c0

(1)

and it is uniquely specified by a polynomial formed by the elementsof its first row, c(x) = c0 + c1x + � � � + cm�1x

m�1, with the leastsignificant coefficient on the left.

A 1-generator QC code has the following form of the generator ma-trix [9]:

G = [ G0 G1 G2 . . . Gp�1 ] (2)

where Gi, i = 0; 1; 2; . . . ; p � 1, are circulants of order m. Letg0(x); g1(x); . . . ; gp�1(x) are the corresponding defining polyno-mials.

A 2-generator QC [m � p; k] codes has the generator matrix of thefollowing form:

G =G00 G01. . . G0;p�1

G10 G11. . . G1;p�1(3)

where Gij are circular matrices, for i = 0, and 1, j = 0; 1; . . . ; p� 1.Similarly, a 3-generator QC [m�p; k] codes has the generator matrix

of the following form:

G =

G00 G01 . . . G0;p�1

G10 G11 . . . G1;p�1

G20 G21 . . . G2;p�1

(4)

whereGij are circular matrices, for i = 0, 1, and 2, j = 0; 1; . . . ; p�1.

III. CONSTRUCTIONS OF 2-GENERATOR QC CODES

Given any positive integer k. If there exist two binary cyclic Ham-ming [2k� 1; 2k�k� 1; 3] codes, then there exist two cyclic simplex[2k � 1; k; 2k�1] codes. Let g1(x) and g2(x) be the generator poly-nomials of these simplex codes, C1 and C2. A binary 2-generator QC[(2k � 1)p; 2k] code can be constructed with the following generatormatrix:

G =G1 G1 . . . G1

G2 G2;1 . . . G2;p�1(5)

where G1 is the circulant matrix defined by the generator polynomialg1(x), G2 is the circulant matrix defined by g2(x), and G2;i is thecirculant matrix defined by xa(i)g2(x), where 0 � a(i) < 2k � 1 , isan integer. The choices of a(i), i = 1; 2; . . . ; p� 1 , are to maximizethe minimum distance to the code.

Let m = 2k � 1, and the distance vector D = (d0; d1; . . . ; dm�1),where di is defined as the distance between the codeword g1(x) inC1 and the codeword xig2(x) in C2. Then a good 2-generator QC[m � p; 2k] code can be obtained by choosing a(i) to maximize itsminimum distance:

d = min(dj + dj+a(1) + � � �+ dj+a(p�1));

where j = 0; 1; . . . ;m � 1.Example 1: n = 7, k = 3. x7�1 = (x+1)(x3+x+1)(x3+x2+

1). So two cyclic simplex [7; 3; 4] codes are defined respectively byg1(x) = x4+x3+x2+1 and g2(x) = x4+x2+x+1. The distancevector D = (2; 4; 4; 6; 2; 2; 4). Let p = 3, an optimal 2-generator QC[21; 6; 8] code can be obtained with a(1) = 1, and a(2) = 2.

For n = 15, k = 4, we get the equation at the bottom of thepage. Two cyclic simplex [15; 4; 8] codes can are defined repectivelyby g1(x) = 7531 and g2(x) = 4657 in octal, with the highest degreeterms on the left. The distance vector is

D = (8; 6; 8; 4; 6; 10; 8; 6; 4; 10; 6; 10; 10; 8; 8):

Forn = 31,k = 5. Two cyclic simplex [31; 5; 16] codes are defined,respectively, by g1(x) = 454761565 and g2(x) = 715750453 inoctal. The distance vector is

D = (12; 12; 16; 16; 20; 12; 16; 20; 16; 16; 20; 16; 16; 16; 16;

12; 16; 20; 16; 12; 12; 12; 16; 12; 20; 16; 16; 16; 12; 20; 12):

For n = 63, k = 6. Two cyclic simplex [63; 6; 32] codes are de-fined respectively by g1(x) = 10305172162267315277 and g2(x) =13745214756551542207 in octal. The distance vector is

D =(32; 32; 24; 40; 32; 32; 40; 32; 32; 32; 32; 32; 32; 24; 32;

32; 32; 32; 24; 32; 40; 32; 32; 32; 32; 32; 40; 32; 32; 32; 32;

24; 32; 32; 32; 32; 32; 32; 32; 32; 24; 32; 32; 32; 32; 32; 32;

32; 32; 24; 24; 24; 32; 24; 40; 32; 32; 24; 32; 40; 32; 32; 32):

Table I lists good binary 2-generator QC codes constructed. In thetable, superscript “o” denotes the code obtained is optimal, “=” denotesthe code meets the best minimum distance in [12].

Many projective two-weight codes are also constructed. For exam-ples, binary 2-generator QC two-weight codes with p = 3; 4; 7 form = 7, and p = 10; 11; 12, and 15 for m = 15 are obtained. In [11],QC two-weight codes are discussed and many codes are constructed.

IV. CONSTRUCTIONS OF 3-GENERATOR QC CODES

Similarly, if there exist three binary cyclic simplex [2k�1; k; 2k�1]codes,C1,C2, andC3, defined by generator polynomials g1(x), g2(x),and g3(x), respectively, then a binary 3-generator QC [(2k � 1)p; 3k]code can be constructed as follows:

G =

G1 G1. . .G1

G2 G2;1. . .G2;p�1

G3 G3;1. . .G3;p�1

(6)

where G1 is the circulant matrix defined by the generator polynomialg1(x), G2 is the circulant matrix defined by g2(x), G3 is the circulantmatrix defined by g3(x), G2;i is the circulant defined by xa(i)g2(x),G3;i is the circulant defined by xb(i)g3(x), where 0 � a(i) < 2k � 1,

x15� 1 = (x+ 1)(x2 + x+ 1)(x4 + x+ 1)(x4 + x

3 + 1)(x4 + x3 + x

2 + x+ 1):

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007 1195

TABLE IGOOD BINARY 2-GENERATOR QC [pm; 2k] CODES

and 0 � b(i) < 2k � 1. The choices of integers a(i) and b(i), i =1; 2; . . . ; p� 1 , are to maximize the minimum distance to the code.

Let m = 2k � 1, and the distance vectors

D12 = (d120 ; d121 ; . . . ; d12m�1)

D13 = (d130 ; d131 ; . . . ; d13m�1)

D23 = (d230 ; d231 ; . . . ; d23m�1)

where d12i is the distance between the codeword g1(x) in C1 and thecodeword xig2(x) in C2, d13i , the distance between the codewordg1(x) in C1 and the codeword xig3(x) in C3, and d23i the distancebetween the codeword g2(x) in C2 and the codeword xig3(x) in C3.Let the distance table D123 be

d1230;0 ; d

1230;1 ; . . . ; d

1230;m�1

d1231;0 ; d

1231;1 ; . . . ; d

1231;m�1

. . .

d123m�1;0; d

123m�1;1; . . . ; d

123m�1;m�1

where d123i;j is the weight of the sum of codewords g1(x) inC1, xig2(x)in C2 and xjg3(x) in C3.

Then a good 3-generator QC [m � p; 3k] code can be obtained bychoosing integers a(i) and b(j) to maximize its minimum distance

d12j + d

12j+a(1) + � � �+ d

12j+a(p�1)

d13j + d

13j+b(1) + � � �+ d

13j+b(p�1)

d23j + d

23j+b(1)�a(1) + � � �+ d

23j+b(p�1)�a(p�1)

d123i;j + d

123i+a(1);j+b(1) + � � �+ d

123i+a(p�1);j+b(p�1)

where i, j = 0; 1; . . . ;m�1, and subscripts are computed modulo m.For k = 5, three cyclic simplex [31; 5; 16] codes are defined by

generator polynomials g1(x) = 535437151, g2(x) = 454761565 andg3(x) = 715750453 in octal. For p = 3, a 3-generator QC [93; 15; 36]code is obtained with a(1) = 1, a(2) = 18, and b(1) = 30, b(2) = 2.This code meets the lower bound on the minimum distance. For thecode constructed in the methods given above, It is often possible toextend the code by adding one or more information digits and parity

check digits. Based on the [93; 15; 36] code, a new 5-generator QC[93; 17; 34] code is obtained. Its generator matrix is shown as follows:

G =

G1 G1 G1

G2 G2;1 G2;2

G3 G3;1 G3;1

1 . . . 1 1 . . . 1 0 . . . 0

1 . . . 1 0 . . . 0 1 . . . 1

:

Further, if three additional parity check digits(one for each 31 bits)are added, a new binary [96; 17; 36] code is obtained. Both [93; 17; 34]and [96; 17; 36] codes improve the lower bound on the minimum dis-tance on binary linear codes.

For k = 7, three cyclic simplex [127; 7; 64] codes are defined bygenerator polynomials

g1(x) = 0017725147351306755331107027625632117050301

g2(x) = 11151734177073051372502674712630155350621

and

g3(x) = 14111773707251275147153042731036267012155

in octal. A 3-generator QC [254; 21; 104] code can be obtained witha(1) = 21, and b(1) = 43. Two additional information digits can beadded to this code by adding two extra rows to the generator matrix,a new 5-generator QC [254; 23; 102] code can be obtained with thefollowing generator matrix:

G =

G1 G1

G2 G2;1

G3 G3;1

1 . . . 1 0 . . . 0

0 . . . 0 1 . . . 1

:

If two additional parity check digits(one for each 127 bits) are added,a new linear [256; 23; 104] code can be obtained. Both [254; 23; 102],and [256; 23; 104] codes improve the lower bound on the minimumdistance on binary linear code.

V. CONCLUSION

In this correspondence, construction methods for 2- and 3-generatorQC codes from cyclic simplex codes are presented. Many good new2-, 3-generator QC codes are found, and new linear codes that improvethe lower bounds on the minimum distance are constructed.

REFERENCES

[1] C. L. Chen and W. W. Peterson, “Some results on quasi-cyclic codes,”Inf. Contr., vol. 15, pp. 407–423, 1969.

[2] E. J. Weldon Jr., “Long quasi-cyclic codes are good,” IEEE Trans. Inf.Theory, vol. IT-13, no. 1, pp. 130–130, Jan. 1970.

[3] T. Kasami, “A Gilbert-Varshamov bound for quasi-cyclic codes of rate1/2,” IEEE Trans. Inf. Theory, vol. IT-20, no. 5, p. 679, Sep. 1974.

[4] S. Ling and P. Solé, “Good self-dual quasi-cyclic codes exist,” IEEETrans. Inf. Theory, vol. 49, no. 4, pp. 1052–1053, Apr. 2003.

[5] H. C. A. van Tilborg, “On quasi-cyclic codes with rate 1=m,” IEEETrans. Inf. Theory, vol. IT-24, no. 5, pp. 628–629, Sep. 1978.

[6] T. A. Gulliver and V. K. Bhargava, “Some best rate 1=p and rate (p�1)=p systematic quasi-cyclic codes,” IEEE Trans. Inf. Theory, vol. 37,no. 3, pp. 552–555, May 1991.

[7] P. Heijnen, H. van Tilborg, T. Verhoeff, and S. Weijs, “Some new bi-nary quasi-cyclic codes,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp.1994–1996, Sep. 1998.

1196 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007

[8] E. Z. Chen, Web Database of Binary QC Codes [Online]. Available:http://www.tec.hkr.se/~chen/research/codes/searchqc2.htm

[9] G. E. Séguin and G. Drolet, “The theory of 1-generator quasi-cycliccodes,” Dept. Elec. Comp. Eng., Royal Military College of Canada,Kingston, ON, Canada, 1990.

[10] A. E. Brouwer, Bounds on the Minimum Distance of Linear Codes[Online]. Available: http://www.win.tue.nl/~aeb/voorlincod.html

[11] Z. C. Eric, “Constructions of quasi-cyclic two-weights codes,” in Proc.10th Int. Workshop on Algebraic and Combinatorial Coding Theory,Zvenigorod, Russia, Sep. 2006.

[12] T. A. Gulliver and V. K. Bhargava, “Two new rate 2=p binary quasi-cyclic codes,” IEEE Trans. Inf. Theory, vol. 40, no. 5, pp. 1667–1668,Sep. 1994.

On the Error Exponents of Improved TangentialSphere Bounds

Moshe Twitto, Student Member, IEEE, and Igal Sason, Member, IEEE

Abstract—The performance of maximum-likelihood (ML) decoded bi-nary linear block codes over the additive white Gaussian noise (AWGN)channel is addressed via the tangential sphere bound (TSB) and two of itsrecent improved versions. The correspondence is focused on the derivationof the error exponents of these bounds. Although it was shown that somerecent improvements of the TSB tighten this bound for finite-length codes,it is demonstrated in this correspondence that their error exponents coin-cide. For an arbitrary ensemble of binary linear block codes, the commonvalue of these error exponents is explicitly expressed in terms of the asymp-totic growth rate of the average distance spectrum.

Index Terms—Block codes, bounds, linear codes, maximum-likelihood(ML) decoding.

I. INTRODUCTION

In recent years, much effort has been put into the derivation of tightperformance bounds on the error probability of linear block codesunder soft-decision maximum-likelihood (ML) decoding. During thelast decade, this research work was stimulated by the introduction ofvarious codes defined on graphs and iterative decoding algorithms,achieving reliable communication at rates close to capacity withfeasible complexity. The remarkable performance of these codes atrates above the channel cutoff rate makes the union bound useless at aportion of the rate region where their performance is most appealing.Hence, tighter performance bounds are required to gain some insighton the performance of these efficient codes. Improved upper and lowerbounds on the error probability of linear codes under ML decoding areaddressed in [12] and references therein, and these bounds are appliedto various codes and ensembles.

The tangential sphere bound (TSB) [9] forms one of the tightestupper bounds on the error probability for ML decoded linear block

Manuscript received March 27, 2006; revised July 16, 2006. The material inthis correspondence was presented in part at the 2006 IEEE 24th Convention ofElectrical and Electronics Engineers in Israel, Eilat, Israel, November 2006.

The authors are with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Technion City, Haifa 32000, Israel (e-mail:tmoshe@tx.technion.ac.il; sason@ee.technion.ac.il).

Communicated by R. J. McEliece, Associate Editor for Coding Theory.Color version of Fig. 3 in this correspondence is available online at http://

ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2006.890725

codes whose transmission takes place over the binary-input additivewhite Gaussian noise (BIAWGN) channel. The TSB was modified bySason and Shamai [10] for the analysis of the bit-error probability oflinear block codes, and was slightly refined by Zangl and Herzog [19].This bound only depends on the distance spectrum of the code, andhence, it can be applied to various codes and ensembles. The TSB fallswithin the class of upper bounds whose derivation relies on the basicinequality

Pr(word error j ccc0) � Pr(word error; yyy 2 R j ccc0) + Pr(yyy =2 R j ccc0)(1)

where ccc0 is the transmitted codeword, yyy denotes the received vector atthe output of the channel, and R designates an arbitrary geometricalregion which can be interpreted as a subset of the observation space.The idea is to use the union bound only for the joint event where thedecoder fails to decode correctly and the received vector fails inside theregionR (i.e., the union bound is used for upper-bounding the first termon the right-hand side (RHS) of (1). The TSB, for example, uses a cir-cular hyper-cone as the regionR. Other upper bounds from this familyare addressed in [12, Sects. III and IV], [18], and references therein. In[15], Yousefi and Khandani prove that among all the volumesR whichpossess some symmetry properties, the circular hyper-cone yields thetightest bound. This finding demonstrates the optimality of the TSBamong a family of bounds associated with geometrical regions whichpossess some symmetry properties, and which are obtained by applyingthe union bound to the first term on the RHS of (1). In [16], Yousefi andKhandani suggest to use Hunter’s bound [8] (an upper bound whichbelongs to the family of second-order Bonferroni-type inequalities [5])instead of the union bound. This modification results in a tighter upperbound, referred to as the added hyper plane (AHP) bound. Yousefi andMehrabian [17] also rely on Hunter’s bound, but implement it in a quitedifferent way in order to obtain an improved tangential sphere bound(ITSB) which solely depends on the distance spectrum of the code. Thetightness of the ITSB and the AHP bound is shown in [16] and [17] forsome binary linear block codes of short block lengths; these boundsslightly outperform the TSB at low singal-to-noise ratio (SNR) values.

An issue which is not addressed analytically in [16] and [17] iswhether the new upper bounds (namely, the AHP bound and the ITSB)provide an improved lower bound on the error exponent as compared tothe error exponent of the TSB. In this correspondence, we address thisquestion, and prove that the error exponents of these improved versionos TSB coincide with the error exponent of the TSB. We note howeverthat the TSB fails to reproduce the random coding error exponent, es-pecially for high-rate linear block codes [9].

This correspondence is organized as follows: The TSB ([9], [10]),the AHP bound [16], and the ITSB [17] are presented as a preliminarymaterial in Section II (some boundary effects, which are not consideredin [16] and [17] are discussed in detail in Section II). In Section III, wederive the error exponents of the ITSB and the AHP bound, and stateour main result. We conclude our discussion in Section IV. Appendicesprovides supplementary details related to the proof of our main result.

II. PRELIMINARIES

We introduce in this section some preliminary material which servesas a preparatory step toward the presentation of the material in Sec-tions II-B and II-C. We also present notation from [1] which is usefulfor our analysis. The reader is referred to [12] and [18] which introducematerial covered in this section. However, in the following presenta-tion, we consider boundary effects which were not taken into accountin the original derivation of the TSB and its recent two improved ver-sions in [7], [9], [16]–[18]. These boundary effects do not have any

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