Complementary Sets, Generalized Reed–Muller Codes, and Power Control for OFDM

808 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 2, FEBRUARY 2007

Also, for 1 � l � m, by (4), (5), (7) and (8)

dH(ul; u) = [n�m(k � 1)� jTlj] + [m(k � 1)� (k � 1)]

�n� (m� 1)(k� 1)� (k � 1)

=n�m(k� 1): (10)

Definition 2.3 together with (9) and (10) show that C is not an m-trace-ability code.

Theorem 2.3: Let C be a linear [n; k; d] MDS code over a finite fieldGF (q) such that n � q + 1. Then, for m � 2, C is an m-traceabilitycode if and only if d > (1� 1=m2)n.

Proof: Immediate from Corollary 2.1 and Theorem 2.2.

III. CONCLUSION

In this correspondence, we have studied the properties of codes inorder to enable a deterministic tracing to the nearest neighbor of an ob-served pirated copy. The main result of the correspondence is that weproved that the known sufficient conditions are not necessary in gen-eral, but are necessary for MDS codes. As future work, we would liketo find necessary and sufficient conditions for general m-traceabilitycodes.

ACKNOWLEDGMENT

The authors would like to thank anonymous referees who providedvery useful comments.

REFERENCES

[1] A. Fiat and M. Naor, “Broadcast encryption,” in Crypto’93 (Lec-ture Notes in Computer Science). Berlin, Heidelberg, New York:Springer-Verlag, 1993, vol. 773, pp. 480–491.

[2] B. Chor, A. Fiat, and M. Naor, “Tracing traitors,” in Crypto’94 (Lec-ture Notes in Computer Science). Berlin, Heidelberg, New York:Springer-Verlag, 1994, vol. 839, pp. 480–491.

[3] B. Chor, A. Fiat, M. Naor, and B. Pinkas, “Tracing traitors,” IEEETrans. Inf. Theory, vol. 46, no. 3, pp. 893–910, May 2000.

[4] A. Fiat and T. Tassa, “Dynamic traitor tracing,” in Crypto’99 (Lec-ture Notes in Computer Science). Berlin, Heidelberg, New York:Springer-Verlag, 1999, vol. 1666, pp. 354–371.

[5] D. Boneh and J. Shaw, “Collusion-secure fingerprinting for digitaldata,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 1897–1905, Sep.1998.

[6] A. Barg, R. Blakely, and G. Kabatiansky, “Digital fingerprintingcodes: Problem statements, constructions, identification of traitors,”IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 852–865, Apr. 2003.

[7] A. Barg and G. Kabatiansky, “A class of I.P.P. codes with efficientidentification,” J. Complexity, vol. 20, no. 2-3, pp. 137–147, Apr./Jun.2004.

[8] H. D. Hollmann, J. J. van Lint, J. P. Linnartz, and L. M. G. M.Tolhuizen, “On codes with the identifiable parent property,” J. Comb.Theory, ser. A, vol. 82, pp. 121–133, 1998.

[9] G. Cohen, S. Encheva, S. Litsyn, and H. G. Schaathun, “Intersectingcodes and separating codes,” Discr. Appl. Math., vol. 128, no. 1, pp.75–83, May 2003.

[10] S. Lin and D. J. Costello, Error Control Coding: Fundamentals andApplications, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2004.

[11] G Cohen and H. G. Schaathun, “Upper bounds on separating codes,”IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1291–1295, Jun. 2004.

[12] J. N. Staddon, D. R. Stinson, and R. Wei, “Combinatorial properties offrameproof and traceability codes,” IEEE Trans. Inf. Theory, vol. 47,no. 3, pp. 1042–1049, Mar. 2001.

[13] D. R. Stinson and R. Wei, “Combinatorial properties and constructionsof traceability schemes and frameproof codes,” SIAM J. Discr. Math.,vol. 11, no. 1, pp. 41–53, 1998.

[14] T. van Trung and S. Martirosyan, “On a class of traceability codes,”Des., Codes Cryptogr., vol. 31, no. 2, pp. 125–132, 2004.

[15] H. Jin, J. Lotspiech, and S. Nusser, “Traitor tracing for prerecorded andrecordable media,” in Proc. ACM Workshop on Digital Rights Manage-ment, Washington, DC, Oct. 2004, pp. 83–90.

[16] H. Jin, J. Lotspiech, and M. Blaum, “Traitor tracing for subscription-based system,” in Proc. Int. Conf. Security and Cryptography, Setubal,Portugal, Aug. 2006, pp. 223–228.

[17] A. Barg, G. Cohen, S. Encheva, G. Kabatiansky, and G. Zemor, “Ahypergraph approach to the identifying parent property: The case ofmultiple parents,” SIAM J. Discr. Math., vol. 14, no. 3, pp. 423–431,2001.

[18] L. A. Bassalygo, M. Burmester, A. Dyachkov, and G. Kabatiansky,“Hash codes,” in Proc. IEEE Symp. Inf. Theory, Ulm, Germany, Jun./Jul. 1997, p. 174.

Complementary Sets, Generalized Reed–Muller Codes,and Power Control for OFDM

Kai-Uwe Schmidt

Abstract—The use of error-correcting codes for tight control ofthe peak-to-mean envelope power ratio (PMEPR) in orthogonal fre-quency-division multiplexing (OFDM) transmission is considered in thiscorrespondence. By generalizing a result by Paterson, it is shown that eachq-phase (q is even) sequence of length 2 lies in a complementary set of size2 , where k is a nonnegative integer that can be easily determined fromthe generalized Boolean function associated with the sequence. For smallk this result provides a reasonably tight bound for the PMEPR of q-phasesequences of length 2 . A new 2 -ary generalization of the classicalReed–Muller code is then used together with the result on complementarysets to derive flexible OFDM coding schemes with low PMEPR. Thesecodes include the codes developed by Davis and Jedwab as a special case.In certain situations the codes in the present correspondence are similar toPaterson’s code constructions and often outperform them.

Index Terms—Code, complementary, correlation, Golay, orthogonal fre-quency-division multiplexing (OFDM), peak-to-mean envelope power ratio(PMEPR), Reed–Muller, sequence, set.

I. INTRODUCTION

In some applications, the advantages of the orthogonal frequency-di-vision multiplexing (OFDM) modulation technique are outweighed bythe typically high peak-to-mean envelope power ratio (PMEPR) of un-coded OFDM signals. Among various approaches to solve this power-control issue, the use of block coding across the subcarriers [7], [8] isone of the more promising concepts [13]. Here the goal is to designerror-correcting codes that contain only codewords with low PMEPR.

Sequences lying in complementary pairs [5], also called Golaysequences, are known to have PMEPR at most two in q-ary phase-shiftkeying (PSK) modulation [14]. In [4] Davis and Jedwab devel-oped a powerful theory linking Golay sequences with generalizedReed–Muller codes. More specifically, it was shown that a family ofbinary Golay sequences of length 2m organizes in m!=2 cosets ofRM2(1;m) inside RM2(2;m), where RM2(r;m) is the Reed–Mullercode of order r and length 2m [9]. Similarly for h > 1[4] identifies

Manuscript received June 4, 2005; revised May 23, 2006. This work was sup-ported in part by the German Science Foundation (DFG) under Grant FI 470/8-1.The material in this correspondence was presented in part at IEEE InternationalSymposium on Information Theory, Adelaide, Australia, September 2005.

The author is with the Communications Laboratory, Dresden University ofTechnology, 01062 Dresden, Germany (e-mail: [email protected]).

Communicated by K. G. Paterson, Associate Editor for Sequences.Digital Object Identifier 10.1109/TIT.2006.889723

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 2, FEBRUARY 2007 809

m!=2 cosets of RM2

(1;m) comprised of polyphase Golay sequencesinside ZRM

2(2;m). Here RM

2(r;m) and ZRM

2(r;m) are gener-

alizations of the classical Reed–Muller code over 2h-ary alphabets. Forsmall m, say m � 5, the union of these cosets yields a powerful codewith good error-correcting properties and strictly bounded PMEPR.

However the rate of this code rapidly tends to zero when the blocklength increases. Therefore Davis and Jedwab proposed [4] to includefurther cosets of RM

2(1;m) in order to increase the code rate at

the cost of a slightly larger PMEPR. While in [4] such cosets havebeen identified with an exhaustive search, a more sophisticated theorywas developed by Paterson in [12]; it was shown that each coset ofRM

2(1;m) inside RM

2(2;m) can be partitioned into complemen-

tary sets of size 2k+1, where k is a nonnegative integer that can beeasily determined from a representative of the coset. Since the PMEPRof each sequence lying in a complementary set of size N has PMEPRat most N , [12] provides an upper bound on the PMEPR of arbitrarysecond-order cosets of RM2 (1;m). This result was then exploited invarious ways to obtain coding schemes for OFDM, which extend thoseproposed in [4].

Several further constructions linking complementary sets andReed–Muller codes have been described in [16], [11], [2]. Althoughthese results often provide better upper bounds on the PMEPR thanthe work in [12], it seems difficult to use them to derive practicablecoding schemes for OFDM.

In this correspondence we generalize the results from [12]. We willestablish a construction of sequences that are contained in higher-ordergeneralized Reed–Muller codes and lie in complementary sets of agiven size. It appears that [12, Th. 12] is a special case of this result. Wewill then relate this construction to a new generalization of the classicalReed–Muller code, which we call the effective-degree Reed–Mullercode. In this way, we derive a number of new flexible OFDM codingschemes with low PMEPR. In contrast to the work in [12], all thesecodes arise in a uniform way from a general framework. Moreover theyoften outperform the coding options presented in [12]. The proposedcodes are unions of cosets of a linear code over 2 that contains ingeneral more codewords than RM2 (1;m), although this linear codeitself is a union of cosets of RM2 (1;m). Compared to the approachesin [4] and [12], this makes our codes more amenable to efficient en-coding and decoding algorithms.

The remainder of this correspondence is organized as follows.In Section II we describe a simplified OFDM model, establish ourmain notation (which essentially follows that in [12]), and presentsome known results from [12]. Section III contains our results oncomplementary sets. In Section IV we introduce the effective-degreeReed–Muller code and derive OFDM codes with low PMEPR. Weclose with a discussion in Section V.

II. PRELIMINARIES

A. The OFDM Coding Problem

We consider an OFDM system with n subcarriers. The trans-mitted OFDM signal corresponding to the codeword CCC =(C0; C1; . . . ; Cn�1) 2 n is the real part of the complex enve-lope, which can be written as

S(CCC)(�) =

n�1

i=0

Ciep�12�(i+�)�; 0 � � < 1

where � is a positive constant. In the following it is assumed that theelements of CCC are taken from a q-ary PSK constellation, i.e., Ci = �c

with � = ep�12�=q and ci 2 q . ThenCCC is a polyphase sequence. This

assumption together with Parseval’s identity implies that the complexenvelope has mean power equal to n. The PMEPR of the codeword CCC(or of the corresponding complex envelope) is then defined to be

PMEPR(CCC)1

nsup

0��<1jS(CCC)(�)j2:

The PMEPR is always less than or equal to n, where the maximumoccurs, for example, if CCC is the all-one word. We aim at constructingcodes C that have error-correcting capabilities and for which the value

maxCCC2C

PMEPR(CCC)

is substantially lower than n.

B. Aperiodic Correlations and Complementary Sets

Given two complex-valued sequencesAAA = (A0; A1 . . . ; An�1) andBBB = (B0; B1 . . . ; Bn�1) of length n, their aperiodic cross-correla-tion at a displacement ` 2 is defined to be

C(AAA;BBB)(`)

n�`�1

i=0

Ai+`B�i 0 � ` < n

n+`�1

i=0

AiB�i�` �n < ` < 0

0 otherwise,

where (�)� denotes complex conjugation. The aperiodic auto-correla-tion of AAA at a displacement ` 2 is defined as

A(AAA)(`) C(AAA;AAA)(`):

Definition 1: A set of N sequences fAAA0; AAA1; . . . ; AAAN�1g is a com-plementary set of size N if

N�1

i=0

A(AAAi)(`) = 0 for each ` 6= 0:

If N = 2, the set is called a complementary pair (or Golay comple-mentary pair) [5] and the sequences therein Golay sequences.

Golay sequences have found applications in many different areas ofsignal processing. The work of Popovic [14], where it was essentiallyproved that polyphase Golay sequences have PMEPR at most 2, mo-tivated the use of such sequences as codewords in OFDM [18], [17],[10], [4]. Paterson [12] generalized these results by proving:

Theorem 2 ([12]): Each polyphase sequence lying in a complemen-tary set of size N has PMEPR at most N .

C. Generalized Boolean Functions, Associated Sequences and TheirCorrelations

A generalized Boolean function f is defined as a mappingf : f0; 1gm ! q . Such a function can be written uniquely in thepolynomial form

f(x0; x1; . . . ; xm�1) =i2f0;1g

ci

m�1

�=0

xi� ; ci 2 q

called the algebraic normal form of f . Sometimes we write f in placeof f(x0; x1; . . . ; xm�1). If ci = 1 for exactly one i and zero otherwise,


then f is called a monomial. Let deg(f) denote the algebraic degreeof f .

A generalized Boolean function may be equally represented by se-quences of length 2m. Therefore suppose 0 � i < 2m has binaryexpansion (i0; i1; . . . ; im�1) such that i = m�1

�=0 i�2� and i� 2

f0; 1g, and write fi = f(i0; i1; . . . ; im�1). We define

(f) (f0; f1; . . . ; f2 �1)

as the q-valued sequence associated with f and

(f) (�f ; �f ; . . . ; �f )

as the polyphase sequence associated with f , where � = ep�12�=q .

In what follows we recall the technique of restricting generalizedBoolean functions and their associated polyphase sequences. This tech-nique was introduced in [12] in order to expand aperiodic correlations,as we shall see in Lemma 3.

Suppose that f : f0; 1gm ! q is a generalized Boolean functionin the variables x0; x1; . . . ; xm�1, and let FFF = (f). Let a list of kindices be given by 0 � j0 < j1 < � � � < jk�1 < m, and writexxx = (xj ; xj ; . . . ; xj ). Let ddd = (d0; d1; . . . ; dk�1) be a binaryword of length k, and let (i0; i1; . . . ; im�1) be the binary expansion of0 � i < 2m. The restricted sequence FFF jxxx=ddd is a sequence of length2m that coincides with FFF at the positions i where ij = d� for each0 � � < k. Otherwise FFF jxxx=ddd is equal to zero. For k = 0 we defineFFF jxxx=ddd FFF .

A sequence that is restricted in k variables comprises 2m � 2m�k

zero entries and 2m�k nonzero entries. Those nonzero entries are deter-mined by a function, which is denoted as f jxxx=ddd and called a restrictedgeneralized Boolean function. This function is a generalized Booleanfunction in m� k variables and is obtained by replacing the variablesxj by d� for all 0 � � < k in the algebraic normal form of f . Therestricted sequenceFFF jxxx=ddd is then recovered by associating a polyphasesequence of length 2m�k with f jxxx=ddd and inserting 2m � 2m�k zerosat the corresponding positions. Similarly to a disjunctive normal formof a Boolean function [9], the original function f can be reconstructedfrom the functions f jxxx=ddd by

f =

ddd2f0;1gf jxxx=ddd

k�1

�=0

xdj (1� xj )(1�d ):

Lemma 3 ([12]): Suppose that f : f0; 1gm ! q is a generalizedBoolean function, and let FFF = (f). Let 0 � j0 < j1 < � � � <jk�1 < m be a list of k indices. Write xxx = (xj ; xj ; . . . ; xj ), andlet ddd; ddd1; ddd2 2 f0; 1gk . Then we have

A(FFF )(`) =ddd

A (FFF jxxx=ddd) (`) +ddd 6=ddd

C (FFF jxxx=ddd ; FFF jxxx=ddd ) (`):

D. A Known Construction of Complementary Pairs

Next we recall a construction of complementary pairs from [12].A quadratic polynomial f over q in the f0; 1g-valued variablesxi ; xi ; . . . ; xi is generally given by

f(xi ; . . . ; xi ) =0�j<k<m

bjkxi xi + a(xi ; . . . ; xi )

where bjk 2 q and a is an affine form over q . With each such apolynomial one can associate a labeled graph, denoted by G(f). Thevertices of this graph are labeled with i0; i1; . . . ; im�1, and the edgebetween vertex ij and vertex ik is labeled with bjk.

Such a graph is called a path in m vertices if q is even and m = 1(then the graph consists of a single vertex) or if q is even, m � 2, andf is of the form

f(xi ; . . . ; xi ) =q

2

m�2

�=0

xi xi + a(xi ; . . . ; xi )

where � is a permutation of f0; 1; . . . ;m� 1g. The indices i�(0) andi�(m�1) are called end vertices of the path. If the path consists of asingle vertex, this vertex is called an end vertex as well.

We are now in a position to quote:

Theorem 4 ([12]): Suppose m > k. Let 0 � j0 < j1 < � � � <jk�1 < m be a list of k indices, write xxx = (xj ; xj ; . . . ; xj ), andlet ddd 2 f0; 1gk . Suppose f : f0; 1gm ! q is a generalized Booleanfunction such that f jxxx=ddd is quadratic andG (f jxxx=ddd) is a path inm�kvertices. WriteFFF = (f) andFFF 0 = (f+(q=2)xa+c

0). ThenFFF jxxx=dddand FFF 0jxxx=ddd form a complementary pair. Here, a is an end vertex of thepath G (f jxxx=ddd) and c0 2 q .

In particular, if k = 0, the preceding theorem identifies (m!=2)qm+1

polyphase sequences lying in complementary pairs [12, Cor. 11], whichgeneralizes the original result by Davis and Jedwab [4, Th. 3] from qbeing a power of 2 to even q.

III. A CONSTRUCTION OF COMPLEMENTARY SETS

In what follows we prove that each polyphase sequence of length2m lies in a complementary set, whose size can be easily determinedby inspecting the generalized Boolean function associated with thesequence.

Theorem 5: Suppose m > k. Let 0 � j0 < j1 < � � � < jk�1 < mbe a list of k indices, and write xxx = (xj ; xj ; . . . ; xj ). Let f :f0; 1gm ! q be a generalized Boolean function such that for eachddd 2 f0; 1gk the restricted function f jxxx=ddd is quadratic and G (f jxxx=ddd)is a path in m� k vertices. Then (f) lies in a complementary set ofsize 2k+1, and the PMEPR of (f) is at most 2k+1.

Proof: Write ddd = (d0; d1; . . . ; dk�1) and ccc =(c0; c1; . . . ; ck�1). Define

FFF cccc = f +q

2

k�1

�=0

c�xj +q

2c0e

where ccc 2 f0; 1gk, c0 2 f0; 1g,

e =

ddd2f0;1gxa

k�1

�=0

xdj (1� xj )(1�d )

and addd is an end vertex of the path G (f jxxx=ddd). We claim that the set

FFF cccc jccc 2 f0; 1gk; c0 2 f0; 1g

which contains (f), is a complementary set of size 2k+1. Toprove this, it has to be shown that the sum of auto-correlations

ccc;c A (FFF cccc ) (`) is zero for each ` 6= 0. We employ Lemma 3 andwrite

ccc;c

A(FFF cccc )(`) =S1 + S2

where

S1 =ccc;c ddd

A (FFF cccc jxxx=ddd) (`)

S2 =ccc;c ddd 6=ddd

C (FFF cccc jxxx=ddd ; FFF cccc jxxx=ddd ) (`):


We first focus on the term S1, which can be written as

S1 =ccc ddd

[A (FFF ccc0jxxx=ddd) (`) + A (FFF ccc1jxxx=ddd) (`)] :

Note that ejxxx=ddd = xa . Thus the restricted functions corresponding toFFF ccc0jxxx=ddd and FFF ccc1jxxx=ddd are of the form

f jxxx=ddd +q

2

k�1

�=0

c�d�

f jxxx=ddd +q

2

k�1

�=0

c�d� +q

2xa

respectively. Notice that the term containing the sum over � is a con-stant occurring in both functions. Hence, by hypothesis and by The-orem 4, FFF ccc0jxxx=ddd and FFF ccc1jxxx=ddd form a complementary pair. It followsthat the inner term of S1 is zero for each ` 6= 0. Thus also S1 itself iszero for each ` 6= 0.

It remains to show that the sum S2 is zero. This part of the prooffollows more or less the same reasoning as the second part of the proofof [12, Th. 12].

We have a number of notes on Theorem 5. If k = 0, Theorem 5applies to (m!=2)qm+1 polyphase sequences lying in complementarypairs. These are exactly those identified by setting k = 0 in Theorem4. For k > 0 Theorem 5 essentially generalizes [12, Th. 12]; if f isconstrained to be a quadratic generalized Boolean function, then The-orem 5 virtually reduces to [12, Th. 12].

The proof of Theorem 5 shows that the sequence (f) lies in a com-plementary set that can be decomposed into 2k complementary pairsidentified by Theorem 4. By reversing this process, the sequence (f)can be constructed by interleaving 2k Golay sequences from Theorem4. However, the sole application of such an interleaving method wouldnot directly admit the construction of sequences corresponding to gen-eralized Boolean functions of a specific degree, which will be requiredto derive flexible coding schemes in Section IV.

We also remark that it cannot be expected that Theorem 5 providestight PMEPR bounds for each individual sequence, especially when kis large. Indeed [16] (in particular, [16, Th. 3.6]) and the recent work[2] contain significant improvements of Theorem 5 in certain situa-tions. However it seems difficult to exploit these results to derive codingschemes that admit efficient encoding and decoding.

In summary, the usefulness of Theorem 5 lies in the fact that it pro-vides a relatively simple method to identify sets of sequences that cor-respond to generalized Boolean functions of a given (preferably low)degree and whose PMEPR is bounded above by a given power of 2.

We close this section with an example for the application ofTheorem 5.

Example 6: We take q = 2 and m = 4. Let f : f0; 1g4 ! 2 begiven by

f(x0; x1; x2; x3) = x0x1x2l + x0x1x3 + x0x2 + x1x3 + x2x3:

By restricting f in x0 (i.e., xxx = (x0)), we obtain the two restrictedfunctions

f jx =0 =x1x3 + x2x3

f jx =1 =x1x2 + x2x3 + x2

which are quadratic and their associated graphs are paths in 3 vertices.Hence, by Theorem 5, the PMEPR of (f) is at most 4. By directcomputation it can be observed that the true PMEPR of (f) is ap-proximately 3:32.

IV. OFDM CODES WITH LOW PMEPR

A. The Effective-Degree Reed–Muller Code

A code of length n over the ring 2 is linear if it is a submodule ofn2

. A coset of a linear code C � n2

is defined to be faaa+ cccjccc 2 Cg,where aaa 2 n

2is a representative of this coset. Despite the fact that a

linear code C defined over a ring does not necessarily have a basis, onecan associate a generator matrix with C such that the codewords of Care all distinct 2 -linear combinations of the rows of this matrix. Forbackground on linear codes over rings we refer to [6] and [1].

In what follows we generalize the classical Reed–Muller codes [9]to linear codes over 2 . We begin with defining the effective degreeof a generalized Boolean function.

Definition 7: Let f : f0; 1gm ! 2 be a generalized Booleanfunction. We define the effective degree of f to be

max0�i<h

deg f mod 2i+1 � i :

For instance, the function f : f0; 1g3 ! 8 given byf = 4x0x1x2+x1 has effective degree equal to 1. Now letF(r;m; h)be the set of all generalized Boolean functions f0; 1gm ! 2 ofeffective degree at most r. A simple counting argument leads to

log2 jF(r;m; h)j =

r

i=0

hm

i+

h�1

i=1

(h� i)m

r + i: (1)

Definition 8: For 0 � r � m we define the effective-degreeReed–Muller code as

ERM(r;m; h) f (f)jf 2 F(r;m; h)g :

It follows that ERM(r;m; h) is a linear code over 2 and, sincethe effective degree and the algebraic degree coincide for h = 1,ERM(r;m; 1) is the classical Reed–Muller code [9]. A generator ma-trix for ERM(r;m; h) has rows corresponding to the words associ-ated with monomials in the variables x0; x1; . . . ; xm�1 of degree atmost r together with 2i times the monomials of degree r + i, wherei = 1; . . . ; h� 1. For example a generator matrix for ERM(0; 3; 3) isgiven by

1 1 1 1 1 1 1 1

0 2 0 2 0 2 0 2

0 0 2 2 0 0 2 2

0 0 0 0 2 2 2 2

0 0 0 4 0 0 0 4

0 0 0 0 0 4 0 4

0 0 0 0 0 0 4 4

1

2x02x12x2

4x0x14x0x24x1x2

:

Now let aaa = (a0; a1; . . . ; an�1) be a word with elements in 2 . TheLee weight of aaa is defined to be

wtL(aaa)

n�1

i=0

minfai; 2h � aig

and its squared Euclidean weight (when the entries of aaa are mappedonto a 2h-ary PSK constellation) is given by

wt2E(aaa)

n�1

i=0

j�a � 1j2

where � = ep�12�=2 . Let dL(aaa; bbb) wtL(aaa � bbb) and d2E(aaa; bbb)

wt2E(aaa�bbb) be the Lee and squared Euclidean distance between aaa; bbb 2n2

, respectively. We shall use the standard notation dL(C) and d2E(C)to refer to the respective minimum distances (taken over all distinct


codewords) of a code C � n2

. The minimum squared Euclideandistance of a code essentially determines the performance of the codewhen employed for transmission over a white Gaussian noise channelat high signal-to-noise ratios.

Theorem 9: We havedL(ERM(r;m; h)) = 2m�r

d2E(ERM(r;m; h)) = 2m�r+2 sin2�

2h:

Proof: Since ERM(r;m; h) is linear, its minimum Lee distanceis equal to the minimum Lee weight of the nonzero codewords. We shallfirst find a lower bound for the minimum Lee weight. It is then shownat the end of the proof that this bound is tight. Since ERM(r;m; 1)is the classical Reed–Muller code, the theorem holds for h = 1 (cf.[9]). This case serves as the anchor for the following induction. Letaaa = (a0; a1; . . . ; an�1) be a nonzero codeword in ERM(r;m; h).For h > 1 let bi = ai mod 2h�1. Then bbb = (b0; b1; . . . ; bn�1) is acodeword in ERM(r;m; h � 1). Since ai 2 fbi; bi + 2h�1g, it holdsminfai; 2

h � aig � minfbi; 2h�1 � big, and therefore, wtL(aaa) �

wtL(bbb) � 2m�r , by induction on h.Now let us prove a lower bound on the minimum squared Euclidean

distance. Again we have to find the minimum of wt2E(aaa) taken over allnonzero words aaa 2 ERM(r;m; h). For any u 2 2 we have

j�u � 1j2 =4 sin2 u�

2h

=4 sin2 wtL(u)�

2h:

For 1 � w � 2h�1 it can be shown that

sin2 w�

2h� w sin2

�

2h:

Let Naaa(w) denote the number of entries in aaa with Lee weight equal tow. Indeed

wt2E(aaa) = 4

2

w=1

Naaa(w) sin2

w�

2h

� 4 sin2�

2h

2

w=1

wNaaa(w) (2)

=4 sin2�

2hwtL(aaa):

It remains to exhibit a codeword in ERM(r;m; h), for which the lowerbounds are tight. Such a word is, for example, the word associated withthe monomial x0x1 . . . xr . This word has Lee weight 2m�r , and sinceit only contains zeros and ones, equality holds in (2).

Next we relate ERM(r;m; h) to the codes RM2 (r;m) andZRM2 (r;m) given in [4]. These codes also generalize the binaryReed–Muller code to linear codes over 2 . We have

RM2 (r;m) � ERM(r;m; h)

where the inclusion is proper if h > 1 and r < m. Hence for h > 1and r < m the code ERM(r;m; h) contains more codewords thanRM2 (r;m), while both codes have minimum Lee distance equal to2m�r . For h � 2

ZRM2 (r + 1;m) � ERM(r;m; h)

which is a proper inclusion if h > 2 and r < m� 1. Hence for h > 2and r < m� 1 the code ERM(r;m; h) contains more codewords thanZRM2 (r + 1;m), while their minimum Lee distances are equal to2m�r .

B. OFDM Code Constructions

We begin with defining a linear code over 2 .Definition 10: For 0 � k < m, 0 � r � k + 1, and h � 1 we

define the code A(k; r;m; h) to be the set of words corresponding tothe set of polynomials

m�k�1

i=0

x�gi(xm�k; . . . ; xm�1) + g(xm�k; . . . ; xm�1)

g0; . . . ; gm�k�1 2 F(r � 1; k; h); g 2 F(r; k; h) :

Notice that A(0; 1;m; h) is equal to the generalized first-orderReed–Muller code RM2 (1;m), described in [4]. It follows fromDefinition 10 that A(k; r;m; h) is a linear code over 2 . Moreover

A(k; r;m; h) � ERM(r;m; h)

and therefore, the minimum distances of A(k; r;m; h) can be lower-bounded with Theorem 9. We remark that, similarly as in the proofof Theorem 9, a particular word in A(k; r;m; h) can be identifiedshowing that the lower bounds are in fact tight. The number of code-words in A(k; r;m; h) is equal to 2s, where

s = (m� k) � log2 jF(r � 1; k; h)j+ log2 jF(r; k; h)j (3)

which can be computed with (1).As an example consider A(1; 0; 3; 3). This code is a linear subcode

of ERM(0; 3; 3) and has a generator matrix

1 1 1 1 1 1 1 1

0 2 0 2 0 2 0 2

0 0 2 2 0 0 2 2

0 0 0 0 2 2 2 2

0 0 0 0 0 4 0 4

0 0 0 0 0 0 4 4

1

2x02x12x24x0x24x1x2

:

Now let R(k;m; h) be the set of words associated with the followingpolynomials over 2

2h�1

ddd2f0;1g

m�k�2

i=0

x� (i)x� (i+1)

k�1

j=0

xd

m�k+j(1� xm�k+j)(1�d ) (4)

where ddd = (d0; d1; . . . ; dk�1) and �ddd are 2k permutations off0; 1; . . . ; m � k � 1g.

Corollary 11: The corresponding polyphase words in the cosets ofA(k; r;m; h) with coset representatives in R(k;m; h) have PMEPRat most 2k+1.

Proof: The corollary is a consequence of Theorem 5 and the fol-lowing observations. By restricting any function corresponding to aword in A(k; r;m; h) in the variables xm�k; . . . ; xm�1, we obtain anaffine function, and by restricting any function associated with a wordinR(k;m; h) in the same variables, we obtain a quadratic polynomial,whose graph is a path of length m� k.

We are now in a position to construct a simple code.

Construction 12: Take a single coset of A(k; r;m; h) that con-tains a word in R(k;m; h). The polyphase versions of the words inthis code have PMEPR at most 2k+1. The code has minimum Lee and


squared Euclidean distance equal to 2m�r and 2m�r+2 sin2 �

2, re-

spectively, and the number of encoded bits per codeword is equal tos = log2 jA(k; r;m; h)j, which is given in (3).

In order to obtain a more elaborate code construction, we prove:

Lemma 13: For m � k > 1 and r > 2 � h the set R(k;m; h)contains

(m� k)!

2

2

(5)

words corresponding to a generalized Boolean function of effective de-gree at most r.

Proof: The set R(k;m; h) contains exactly [(m � k)!=2]2

words, all having effective degree at most k+3�h. Hence the lemmais true for r � k + 3 � h. It is also clear that the expression in (4)has algebraic degree at least 2, so the effective degree is at least 3� h.Now suppose that 3� h � r < k + 3� h, and write ` = r + h� 3,where 0 � ` < k. By factoring out terms in the outer sum in (4), itcan be verified that, if

�(d ;...;d ;d ;...;d ) = �(d ;...;d ;1�d ;...;1�d )

then (4) is independent of the variables xm�k+`; . . . ; xm�1 and, there-fore, has effective degree at most r = `+3�h. This leaves the choiceof 2` = 2r+h�3 permutations of the symbols f0; 1; . . . ;m � k � 1gthat are distinct under reversal (e.g., all permutations satisfying �(0) <�(m�k�1)) to obtain distinct words inR(k;m; h) with effective de-gree at most `+3�h. This leads in total to the number given in (5).

Construction 14: Suppose m � k > 1. Let 2 � r � k + 2 whenh = 1 and 1 � r � k+1 when h > 1. Write r0 = minfr; k+1g. Let2t be the largest power of 2 not exceeding (5). Now take the union of 2t

distinct cosets ofA(k; r0; m; h), each containing a word inR(k;m; h)with effective degree at most r. The PMEPR of the correspondingpolyphase words in this code is at most 2k+1, and one can encode s+ tbits, where s = log2 jA(k; r0;m; h)j. Since the code is a subcodeof ERM(r;m; h), its minimum Lee and squared Euclidean distanceis at least 2m�r and 2m�r+2 sin2 �

2, respectively. These are tight

bounds if r = r0.We remark that, when k = 0, Construction 14 essentially restates

the construction by Davis and Jedwab [4]. A list of coding optionshaving PMEPR at most 4 and at most 8 is compiled in Tables I andII, respectively. The quantities dL and d2E indicate lower bounds forthe minimum Lee and the minimum squared Euclidean distance of thecodes, respectively. The code with rate R1 = s=2m is obtained withConstruction 12, and the code with rate R2 = (s+ t)=2m arises fromConstruction 14. Notice that our definition of the code rate differs fromthe common one log2 jCj=2m. The present definition has the advan-tage that it allows a fair comparison of codes over different alphabetson the basis of code rate and minimum squared Euclidean distance.

Finally, we wish to sketch how the proposed codes can be generallyencoded and decoded. Encoding of the code A is straightforward byusing a generator matrix for A. Encoding of a union of cosets of Acan be performed by using the information symbols partly to encode aword from A and partly to select a coset representative from a storedlist. For decoding one needs to have an efficient algorithm to decodethe linear code A. This already provides a decoder for the codes fromConstruction 12. Then codes from Construction 14 can be decoded byapplying the supercode decoding method, as described in [3] and [4].Such a concept involves subtracting all possible coset representativesfrom the received word in turn, and passing the resulting words to adecoder for the code A. Among those decoder outputs the word that isclosest to the received word determines the final decoding result.

TABLE ICODING OPTIONS WITH PMEPR AT MOST 4

TABLE IICODING OPTIONS WITH PMEPR AT MOST 8

V. DISCUSSION AND RELATIONS TO PREVIOUS CONSTRUCTIONS

It can be observed that QPSK (quaternary PSK) codes are alwaysbetter than BPSK (binary PSK) codes, i.e., we can always construct aQPSK code with higher code rate and the same minimum Euclideandistance as a BPSK code. By moving to larger alphabets, the code ratecan be increased further, but only at the cost of a smaller minimumEuclidean distance.

It should be noted that Corollary 11 and the arising code construc-tions do not exploit Theorem 5 in the most general way. The general-ized Boolean functions corresponding to the words in the cosets identi-fied in Corollary 11 are characterized by the property that by restrictingthe functions in the variables xm�k; . . . ; xm�1, we obtain quadraticfunctions whose graphs are paths in the vertices 0; . . . ;m � k � 1.In order to increase the size of the codes in Constructions 12 and 14,we can, according to Theorem 5, apply any permutation to the m vari-ables in the functions corresponding to the codewords (instead of onlyto a fixed set of m � k variables). This, however, has the unwantedeffect that some codewords are generated more than once. Such an ap-proach, coupled with rather complicated techniques to remove multiplecodewords, has been used in [12], where the functions are constrainedto have quadratic degree. Our approach has the advantage that thesedifficulties are avoided. Moreover, compared to the concept in [12], it


allows us to construct our codes as unions of relatively few cosets of arelatively large linear code, which presumably simplifies the decodingprocess. The penalty of this simplification is a loss of at most log

2

m

k

encodable information bits (since, instead of m

kpossible index sets,

we choose just one set of m � k indices that form the vertices of thepaths in the graphs of the restricted functions). This loss is moderatefor typical choices of m and k.

Finally we wish to compare the codes arising from Construction 14with those in [4] and [12]. The codes in the latter references are con-tained in RM

2(2;m) or ZRM

2(2;m), which ensures a minimum

Lee distance of at least 2m�2 or 2m�1, respectively. So we comparethese codes with codes arising from Construction 14 having the samelower bound on the minimum Lee distance, i.e., we let r 2 f1; 2g. Alsowe let k = 1, which covers the majority of the codes in [4] and [12]having PMEPR greater than 2.

For h = 1, r = 2, and m � 3 the code from Construction 14 canbe used to encode

blog2(m� 1)!c+ 2m� 1

bits. This yields 13 and 17 bits for m = 5 and m = 6, respectively.These values should be compared with 11 and 17 bits in [12, Table I],which suggests that for h = 1 and for small m our construction isslightly stronger than that in [12]. However for h = 1 and m � 8 itwas stated in [12] that the number of encoded bits of the codes in [12]is equal to blog

2m!c + 2m � 2. Hence for large m the binary code

from [12] allows to encode either blog2mc or blog

2mc � 1 bits more

than a comparable code arising from Construction 14. We arrive at asimilar conclusion for h = 2 and r = 1. For m � 3 we can encode

blog2(m� 1)!c+ 3m

bits, which is, compared to a code in [12] with the same minimum dis-tance, slightly larger for smallm and is either blog

2mc or blog

2mc�1

bits less for m � 8.For h � 2, r = 2, and m � 3 Construction 14 yields a code, which

can be used to encode

b2 � log2(m� 1)!c+ 2hm� 2

bits. When m � 4, the number of encoded bits for a comparable codein [12] is equal to blog

2m!c+2hm�2. This is blog

2m!�2 � log

2mc

or blog2m!�2 � log

2mc+1 bits less than the code from Construction

14. Similar results can be established for h > 2 and r = 1.In summary, except for large m in the cases (h; r) = (1; 2) and

(h; r) = (2; 1), the codes from Construction 14 outperform codingschemes proposed in [12].

Based on exhaustive computational search, [4] reports codes thatoutperform the codes given in the first and the third row of Table I byone encoded information bit and the codes in the seventh and ninth rowof Table I by two encoded information bits. These observations can bepartly explained using a variety of individual theorems from [15], [12],[16], [2] and show that stronger constructions are possible in some situ-ations. However the description of such codes (and therefore encodingand decoding) tends to be unwieldy.

ACKNOWLEDGMENT

The author wishes to thank the associate editor and the anonymousreferees for valuable comments which greatly improved the presenta-tion of this correspondence.

REFERENCES

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Complementary Sets, Generalized Reed–Muller Codes, and Power Control for OFDM