IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

~

801

DC-Constrained Codes from Hadamard Matrices

Alexander M. Barg and Simon N. Litsyn

Abstract -The construction of balanced error-correcting codes with distance close to half of the block length and bounded running digital sum is considered. Use of these codes in cascade constructions allows us to obtain a number of classes of dc-con- strained codes of various lengths. We also present a construc- tion based on simple properties of quadratic residues, generaliz- ing the Paley construction of Hadamard matrices. Index Terms-Running digital sum, Legendre symbol, incom-

plete exponential sum.

I. INTRODUCTION GREAT NUMBER of recent papers 131, [51, [71, A [9]-[12], [15], [161, [21] have been devoted to the

study of construction techniques and existence bounds for binary dc-free (dc-constrained) error-correcting codes. The problem of constructing such codes is motivated by the properties of magnetic recording channels and of fiber-optic digital transmission systems. Error correction is necessary for the reliable operation of these systems, and information transmission channels in these systems do not accept zero- and low-frequency components of the signal. Once we consider a discrete sequence in place of a continuous signal, the latter requirement describes se- quences with bounded running digital sum (RDS) (see [15], [211). Roughly speaking, this means that any initial segment of an arbitrary code word contains as many zeros as ones.

Another limitation usually imposed on line codes is that they be self-clocking, meaning that it is possible to derive the clock signal from the data itself. To provide this possibility, it is necessary that code words do not contain long sequences of symbols of like polarity, for this would cause loss of synchronization. This requirement is partially fulfilled if we consider codes with bounded RDS, for then the maximum runlength 1 does not exceed twice the maximum RDS.

Manuscript received September 24, 1990, revised January 9, 1991. This work was presented in part at the 2nd International Workshop on Algebraic and Combinatorial Coding Theory, Leningrad, U.S.S.R., September 1990, and at the 5th International Workshop on Convolu- tional Codes; Multi-User Communication, Moscow, U.S.S.R., January 1991.

A. M. Barg is with the Institute for Problems of Information Transmission, Ermolovoy 19, Moscow GSP-4, 101447, U.S.S.R. abarg@ippi.msk.su.

S. N. Litsyn is at c/o G. D. Cohen, ENST, 46 rue Barrault, 75634 Paris cedex 13, France.

IEEE Log Number 9143290.

Finally, to increase the recording density on magnetic media it is desirable to satisfy a limitation on the mini- mum number b of consecutive zeros between any two neighboring ones. Usually, line codes are described by the triple of parameters (b , l ,C) , where C is the maximum RDS of a code. In the present paper we do not consider the problem of constructing codes with given minimum runlength b (the interested reader is referred to [16]). The codes that we propose, can therefore be described by the triple (0,2C, C) and in this sense coincide with the codes of A. Calderbank et al. [7].

Our paper is organized as follows. In Section 11, we give the definitions of balanced, dc-constrained and dc- free codes and provide the exact formulation of our problem. Essentially, our aim is to investigate constraints on the size of balanced codes with given RDS and Ham- ming distance or, vice versa, to obtain bounds for the RDS of codes with known size and distance. It is evident that any code becomes constant weight (for example, balanced) after adding a tail of appropriate weight to each codeword. Therefore, the main restriction that forms the difference between our problem and conventional problems of coding theory is that on the value of RDS.

In order to construct a balanced dc-constrained (BDCC) code from a balanced one, a natural way is to try to permute the code coordinates in order to minimize the RDS. This leads to a nonconstructive lower bound for the size of BDCC codes, presented in Section 111. Applying the asymptotic expression for the total number of dc-free sequences [23], [9], we readily obtain an asymptotic lower bound for the size of dc-constrained codes. However, investigating asymptotics was not the main purpose of the present paper (see [9], [lo], [161). In the same section, we consider several examples of the application of our bound to short codes (such as Hadamard, Golay, and the like), which allows us to state the existence of a number of codes with very small RDS, listed in Tables 1-111. In Section IV, we estimate the maximum RDS of a class of very good codes-the Hadamard codes, obtaining esti- mates better than those of [7]. As in [5], [7], we substitute these codes as inner codes into a multilevel construction, resulting in a number of families of BDCC codes with great minimum distance. Similar to the weight spectrum, we define the RDS-spectrum of a dc-constrained code. An element of this spectrum is the number of sequences

001 8-9448/9 1 /OSOO-OSO 1 $01 .OO 0 1991 IEEE

lossless of finite order can be viewed as “deterministic with bounded delay.” Fig. 2. Example for bound of Theorem 5.

C . . ” C. ._

802 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

with the same maximum RDS. We list the RDS-spectra of several short Hadamard codes.

Our main result (Section V) is a construction of dc-con- strained codes generalizing the Hadamard codes. Namely, we construct an infinite family of codes of distance slightly less than half of the block length n and bounded RDS. Of course, the codes may again be used in multilevel con- structions.

11. DEFINITIONS Up to now there is no standard terminology in the

theory of balanced codes, therefore our definitions may slightly differ from the definitions of other authors. Nev- ertheless, the interconnection of our definitions with those of other papers is evident.

Let A [ q , n, M , d ] be a q-ary code (in case of q = 2 we sometimes write A [ n , M , d ] ) of length n, containing M words and having minimum distance d . For a codeword a=(a , , ,a , , . - . ,a , - , ) of a binary code A we define its RDS (sometimes called the accumulated charge) at the moment k , 0 I k I n - 1, as

I

Define also the maximum running digital sum (MRDS) of the word a and of the code A as

r ( a ) = max S k ( a ) , O < k < n - l

r ( A ) = max r ( a ) . ( 2 ) a E A

A code A is called balanced if, for every a E A , S,- ,(a) = 0 ( n even). We call a balanced code A dc-con- strained if

r ( A ) s c ( n ) , (3) where c(n) < n is some function in n , and dc-free if c(n) is a constant. Note that the original definition of dc-free sequences differs from our definition. In [15], [21] it is shown that a signal has zero-average power content at zero frequency iff the following condition holds. Consider a semiinfinite sequence x of transmitted data. Then x is dc-free if r(x)=sUpk,OSk(x) is bounded above by a constant. Our definition coincides with this one because within each codeword a the RDS of a transmitted se- quence is not more than a constant and it becomes zero after every n transmitted symbols.

Finally in connection with (3) we would like to point out that the maximum length 1 of a series of identical symbols in a codeword is closely connected with the MRDS for it is clear that 1 I 2r(A).

A quite natural approach to the construction of dc-con- strained codes is the multilevel one. Take a [ 2 k , L , k l binary constant weight code C and concatenate it with a nonbinary [ Q, N , M , D ] code. The resulting concatenated code A is obviously balanced and its MRDS is less than or equal than k . Now consider the case where k is a constant and N +w. Then the code A is dc-free. This is,

more or less, the main idea of the papers [5] and [7]. In particular, in [7] , C is a Hadamard code. For this case, we prove the estimate r(C) I k to be excessively rough. In fact, we show r ( C ) ~ h I n n + 1 , so the code C itself appears to be dc-constrained. Of course, it may be once again substituted into the multilevel construction.

111. EXISTENCE BOUND Suppose A is a balanced code. Then permuting its

coordinates, we might try to lessen its MRDS. The bound (4) appears as a result of averaging over all n! coordinate permutations. It provides the theoretical limits for the BDCC codes.

Denote by

M ( n , c) = I{a E Filr( a ) 5 c, s,-*( a ) = o}I, the number of balanced vectors of length n with given MRDS.

Statement I : Suppose there exists a balanced code A [ n , M , , d ] . Then there exists a BDCC code B [ n , M B , d ] with r ( B ) I c, c 2 1 , and

Proof: Consider n! codes Ai obtained from A after all possible permutations of coordinates. For all i, 0 I i I n! , the minimum distance d ( A i ) = d ( A ) . Every code vec- tor of weight n / 2 will be permuted into all possible vectors of weight n / 2 and every vector of weight n / 2 will be obtained from each code vector ( ~ 2 / 2 ! ) ~ times. In the union of the codes Ai, each vector is encountered M,(m/2!I2 times. All the vectors with MRDS less or equal than c are encountered L, = M ( n , c)M,(m / 2 ! ) , times. So there are n! codes Ai with distance d contain- ing L, dc-constrained vectors. Hence there exists a code

0

To obtain an asymptotic form of the bound (41, we need an asymptotic expression for M(n,c ) . The exact formula for this number appears in [23, Section 211 as a result of consideration of random walk in an interval. An easy corollary of this formula is the following expression

B of size MB 2 L , / n ! , which coincides with (4).

= )). ( 5 )

This expression has been also obtained in [9] . From (4) and ( 5 ) we have the following asymptotic inequality for the rate of the code B.

Corollary I : Suppose there exists a sequence of bal- anced codes A,[n,M,,d, l and R, = l i m , ~ ~ l o g 2 1 M ~ l / n . Then there exists a sequence of BDCC codes B , [ n , M B , d , ] with r(B) I c, c L 1 , and

T R B 2 RA + log2 COS - n+m. ( 6 )

2 c + 2 ’ Another simple observation is that for a self-comple-

mentary code A , when computing MB, we must take the

m , I

BARG AND LITSYN: DC-CONSTRAINED CODES FROM HADAMARD MATRICES

r ( A , ) = max. k,l€F,

803

1 + E x ( l - j ) + A k l j = o

TABLE I LENGTH = 24, MINIMUM DISTANCE = 8, MA = 2576

r ( B ) 1 2 3 4 5 6 7 8 9-12

ME 4 338 1196 1918 2320 2496 2556 2574 2576

TABLE 11-A LENGTH = 16, MINIMUM DISTANCE = 4, MA = 870

r ( B ) 1 2 3 4 5 6 7 8

M, 18 296 626 796 854 868 870 870

TABLE 11-B LENGTH = 16, MINIMUM DISTANCE = 4, M A = 1170

MB 24 398 838 1070 1150 1168 1170 1170

TABLE 111 LENGTH = 16, MINIMUM DISTANCE 8, MA = 30

r ( B ) 1 2 3 4 5 6 7 8

ME 2 12 22 28 30 30 30 30

least euen integer greater than or equal to the right-hand side of (4). For example, taking all vectors of weight n/2 in a Hadamard code, we obtain Corollary 2.

Corollary 2: Suppose there exists a Hadamard matrix of dimension n. Then for all c I n / 2 there exists a BDCC code B[n, MB, n /2] with r ( B ) I c and

M ( n, c)( 2n - 2) -( ( n 7 2 ) ). where ( x ) means rounding up to the least even integer greater than or equal to x .

For example, taking n = 8, we obtain M(8,4)= 70, M(8,3) = 68, M(8,2) = 54, M(8,l) = 16. Hence we have M B 2 1 4 for c = 3 and 4, M B 2 1 2 for c = 2 and MB 2 4 for c = 1. Similarly, starting from 2576 words of weight 12 of the [24,4096,8] Golay code, we obtain BDCC codes listed in Table I. It is worth mentioning that we can transmit 11 information bits with the code B of length 24, distance 8, and MRDS r ( B ) = 5 (compare with [3]). Start- ing from the code A,[ l6,870,4] (subcode of the Hamming code) we obtain BDCC codes listed in Table 11-A. We mention here the Hamming subcode A,, because it ad- mits a very simple decoding. Observe that there exists a much better code A, with the parameters [16,1170,4] (see [6]), which results in the BDCC codes listed in Table 11-B. Our last example here is a [16,30,8]-code, the Hadamard code with the all-zero and all-one codewords removed. Applying Corollary 2, we conclude that there exist BDCC codes with parameters listed in Table 111.

IV. DC-CONSTRAINED CODES FROM HADAMARD MATRICES

The Hadamard matrices are widely applied in different areas of discrete mathematics and also in communication theory, image encoding, etc. It is known [19, Section 2.31

11 ‘I I - - - A

lossless of finite order can be viewed as “deterministic with bounded delay.”

C . . ” C. ._ . . d

that they lead to optimal binary codes, the so-called Hadamard codes. In this section, we examine the value of the MRDS of these codes. It appears that for a prime p with n = p + 1 divisible by 4 and a Hadamard code of length n one can prove r (A) - fi In n instead of a previ- ously known estimate r ( A ) =: n.

Suppose p is a prime with 4lp + 1 and x : F, + { - 1,0,1) is the Legendre symbol. Recall that x(0 ) = 0 and x ( a ) = 1 (resp. - 1) if the equation x 2 = a has (resp. has not) a solution modulo p. The properties of the Legendre sym- bols are discussed in many textbooks of elementary num- ber theory (see, e.g., [14]). Put n = p + 1 and consider the Hadamard matrix of Paley type [191

Here J, is the p X pJacobstha1 matrix with the entries j,, = x ( u - U) and Ip is the identity matrix. It is known that when a runs over Fp*, x ( a ) assumes the values 1 and - 1 equally often. We deduce that the Hamming weight of each but the first row of H, (after changing + 1 to 0 and - 1 to 1) is exactly n /2. Consider p + 1 rows of H,, and their negations. By definition, they form the Hadamard code of Paley type with the parameters [n = p + 1,2p + 2, n /21. Removing from the code the all-cone and all-zero words, we obtain the balanced code A, with the parameters [n = p + 1,2p, ( p + 11/21. Consider the MRDS of this code. Clearly, this problem reduces to the estimation of the incomplete exponential sum

I

(see (9) in the beginning of Section V). The well-known Vinogradov-Polya inequality [18, ch. 51 gives

r , s f i I n p . ( 7)

From (2) and the code definition we obtain I k I

where if k 2 1 ,

A k l = ( i l ’ i f k < l .

Note that we have considered only codewords that are the rows of H,,. It is clear that the negations of these code- words have the same RDS as the words themselves.

Statement 2: For arbitrary prime p with 41p + 1, the balanced code AP[ p + 1,2p, (p + 1)/2] has the MRDS

r ( ~ ~ ) I fi In p + 1. (8)

It follows from [2] that there exist infinitely many values of p, for which the order of magnitude of r(A,) in (8)

Fig. 2. Example for bound of Theorem 5.

804 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

TABLE IV PARAMETERS OF SHORT BDCC CODES OF HADAMARD TYPE

n 8 12 14 18 20 24 30 32 38 42 44 48 54 60 62 d 4 6 6 8 10 12 14 16 18 20 22 24 26 30 30 r ( A p ) 3 4 4 4 5 6 6 7 8 8 8 9 1 0 1 0 8

n 68 72 74 80 84 90 98 102 104 108 110 114 128 d 34 36 36 40 42 44 48 50 52 54 54 56 64 r ( A p ) 10 11 10 11 10 12 12 14 11 12 12 12 12

TABLE V RDS-SPECTRA OF CODES OF HADAMARD TYPE

i 1 2 3 n = 8 p(i) 2 10 2

i 2 3 4 n = 1 2 q(i) 8 10 4

I 3 4 5 6

i 3 4 5 6 7

I 4 5 6 7 8 9 n = 4 8 p( i ) 4 26 34 20 8 1

n = 2 4 p(i) 16 20 8 2

n = 3 2 q(i) 4 28 18 10 2

can be lower bounded as fi In In p. Then it follows from [20] that for these values of p we have in (8) r ( A , ) x G l n l n p . The latter result was proved under the as- sumption of the Riemann hypothesis for zeros of Dirich- let functions. For many values of p the true value of r(A,) is considerably smaller than the estimate (7). Table IV shows parameters of the balanced codes A , for p i 127. We considered all primes p i 1 2 7 (and not only “Hadamard” primes with 41p + 1). The code construction is for all cases one and the same and the distance of the codes of length not a multiple of 4 is slightly less than n /2 .

We define the RDS-spectrum { p , , 1 I i 5 n) of a dc-con- strained code A [ n , M , dl , where p i = IIa E Alr(a) = ill. It is clear that CIi<)p, = M , and removing the words a E A with r ( a ) > i, one obtains the code A’[n, M ‘ = Cf=,,pi, d ‘ 2 d ] with r(A’) I i. In Table V we list the RDS-spectra of some codes from Table IV. It is easy to obtain an asymptotic expression for the RDS-spectrum of a random code, as will be discussed elsewhere.

For arbitrary Hadamard matrices H,, (that is, for n not necessarily equal to a prime plus one) it is possible to improve the dc constraint, removing words U with large values of r (a) from the code. The columns of an arbitrary Hadamard matrix can be ordered in such a way that the first row is the all-one vector and the second one is of the form 1”/20”/2. Then removing the first row (i.e., decreas- ing the code size by 2), we evidently obtain r(A’) = n / 2 , and removing the first two rows, we obtain r (A”) = n /4 . In fact, two arbitrary rows of H,, intersect in n / 4 ones. We see that codes are balanced not only over the length n but also over the length n / 2 , hence r ( A 9 is actually n / 4 .

Use of the Hadamard codes as factors in the Kronecker product [19, Section 18.21 leads to the following balanced codes.

Statement 3: Suppose there exist two Hadamard matri- ces of orders n and N , respectively, and p is a prime with

41p + 1. Then there exist BDCC codes with the following parameters:

1 ) length ( p + 1)N, size 2Np, distance [ ( p + 1)/2]N, and MRDS I fi In p + 1;

2) length nN, size (2n -4)N, distance nN/4 , and MRDS n / 4 .

The parameters of these codes are better than those from Section I1 of [71.

It is clear that all codes from Statement 3 can be split into complementary pairs. Therefore taking these codes as inner codes, we can construct generalized concate- nated codes of second-order [26] with two outer codes. For the codes from Statement 3 0 ) starting from a binary code C,[2, L , M,, 2 d ] and a code C,[Np, L, M,, 2 2d1, we obtain

Statement 4: Suppose there exists a Hadamard matrix of order N and p is a prime with 41p+l. Then there exist BDCC codes with the following parameters: length ( p + 1)NL, size M,M,, distance d ( p + 1)N, and MRDS f i l n p - t l .

In particular, taking a Reed-Solomon code C , (that is N is a degree of p), we have M2 = (NP)~-’~+~.

For codes from Statement 3(2), starting from a binary code C1[2, L , M,, d ] and a code C,[(n - 2)N, L , M2,2d] , we obtain

Statement 5: Suppose there exist two Hadamard matri- ces of orders n and N , respectively. Then there exist BDCC codes with the following parameters: length nNL, size M I M,, distance dnN, and MRDS n /4 .

The codes from Statements 4 and 5 also improve the corresponding results of [71.

In conclusion, we mention a corollary of Statement 3(1). Let c1 be a constant and consider the case p 2 c1 , N +W. Then the statement proves the existence of the dc-free [ L = ( p + 1)N, M = 2Np, L / 2 ] code B, with r (B, ) I c = fi In c, + 1. We see that the size of the code B, satisfies

M = 2 L 1-- i P : l i

Therefore, for any given E > 0 there exists a sequence of balanced [ L , M , L / 2 ] codes with r(B,) i c and M =

2L(1- E ) . Of course, c + w as E -+ 0. On the other hand, the Plotkin bound implies that for any error-correcting [ L , M , L / 2 ] code of even distance we have M < 2 L. We have just shown that there exist error-correcting dc-free balanced codes with distance equal to half of the block

,U, ,,,I

BARG AND LITSYN: DC-CONSTRAINED CODES FROM HADAMARD MATRICES 805

length that are almost as good as any error-correcting codes.

V. DC-CONSTRAINED CODES FROM

LEGENDRE SYMBOLS We present a family of BDCC codes of size 2pm and

with MRDS - m f i Inp with p a prime, of which the Hadamard codes of Section IV are a particular case (m = 1). We start from a slightly different description of Hadamard codes.

We start from an equivalent description of the Hadamard code of Paley type. Suppose {0,1, . . , p - l} are residues modulo p and 6 E Fp is a fixed residue. Consider a vector a of dimension p

a = {X(t) ,X(1+ 0 9 . . ' ,X( (P -1) + 6 ) ) . (9) When 6 runs over F', we obtain p vectors at. The matrix with the rows a* coincides up to a permutation of rows with the Jacobsthal matrix Jp.

This representation of the Hadamard code leads to the following generalization. Pick some m, 2 I m < p and consider the set q l ( p , m), containing all monic square- free polynomials f E Fp[ XI with 1 I deg f I m. Let f? be an arbitrary quadratic nonresidue modulo p and consider also the set 9 , (p , m) = {Pflf E 9 , ( p , m)). It follows from [4], [17] that 9 ( p , m) = g1(p , m>u 9 , (p , m) has the cardinality 2p". Consider the set of ternary vectors

d( p , m) = {di) E { - 1,0, 1jPla(') = (x( f(')(O)),

x( f "'( 1)) , * * 9 x( f "'( P - 1))) 7

1 5 i I 2p").

In order to transform sd(p,m) into a set of binary vectors, change the zeros, corresponding to the roots of the polynomials, according to the following rule: change the first zero in each vector to a symbol that leads to the reduction of the RDS and change all other zeros to 1 and -1 in turn. Putting it in more exact terms, for all i, 1 I i I 2pm, starting from a(') E d ( p , m), form the bi- nary vector b(') according to the following rule. Suppose a vector a(') has mi zeros j = j , < j , < - * * < j m L I m. Then let

bj')=aY), if aj ')#O, 0111p-1; j - 1

(-I) ' , if C a $ ) 2 0 ,

(-I) '+', if a $ ) < ~ .

k = O j - 1

" 1 k = O

b(') =

Note that this substitution may increase the RDS by not more than 1. Again denote symbol - 1 by 1 and + 1 by 0. We obtain the set L@(p,m) of vectors, which are again denoted by b"). Let

S ( P ) m ) = f : l < d e g f < m max 1 c x ( f ( 4 ) l

be the maximum modulus of the complete exponential

sum. It is clear that max, E B(P,m) S p - ,(b) = S ( p , m). Therefore, to balance the code, it is sufficient to add a tail with the necessary number of zeros and ones to each codeword. Let n1 = p + S(p, m). For all i, 1 I i I 2pm, starting from b"), form the vector c( * ) af (even) dimension n = n1 +(n l mod2), setting cjl)= bj'), for 0 I 1 I p - 1;

P - 1

r = O

P - 1

1, if b ( " 2 0 ,

for p I 1 I p + Sp- l (b(r ) ) -1;

0, if b ( ' k 0, I 1 = 0

cy) =

cj t )=((- l ) '+1)/2, forp+S,-,(b(r)) 111n-I.

In other words, we add to the vector b(r)Sp-l(b(')) equal symbols that balance it. The remaining part of the code- word is formed by the alternating 0 and 1 sequence. It is clear that this procedure for even n results in a balanced vector d').

Now consider the set of vectors k ( p , m) = {d'), 1 I i I 2pm}. This is the code we aimed to construct. In order to estimate its parameters we need the following lemma (for the proof see the Appendix). Lemma I: Suppose f is a polynomial of degree 1 I

m s p - 1 . Then

I c x ( f W I < m m n P +(m - 1 ) f i U € U

= m f i In P( 1 + 4 1 ) ) 7 (10) where U = {0,1,- - . , b - 1) c F, is a set of consecutive residues.

In order to estimate the distance of the code 8, con- sider two code vectors @ and d2), obtained from the polynomials fl(x) and f 2 ( x ) . It is clear that d(c(", d2)) 2 d(b('), b(,)) and

-2d(b"',b'2') = c X(fl(X))X(f2(X)) x E F,,

= c X(fl(X)f ,W) x E F,,

1(degf,+degf,-1)fi (11) by the Weil inequality. Taking into consideration that deg f l + deg f, I 2m and that f l and f2 both have in F, not more than m zeros, we obtain the following state- ment.

Statement 6: For arbitrary prime p and 2 I m < p, the set of vectors k ( p , m) = {d'), 1 I i I 2pm} forms a bal- anced code with the following parameters

length

size M = 2p"

12 I p +(m - 1)fi

distance

lossless of finite order can be viewed as "deterministic with bounded delay."

Fig. 2. Example for bound of Theorem 5. C . . " C. ._

h06 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 31, NO. 3, MAY 1991

TABLE VI PARAMETERS OF CODES FROM SECTION V ( r = 2)

Code C[14,338,2], r ( C ) = 6

Code C[18,578,41, r ( C ) = 7

i 1 2 3 4 5 6 pi 16 136 114 58 12 2

i 1 2 3 4 5 6 7 p , 2 158 226 126 44 20 2

i 1 2 3 4 5 6 7 8 p , 8 152 256 194 78 24 8 2

i 2 3 4 5 6 7 8 p , 194 324 254 174 82 26 4

i 2 3 4 5 6 7 8 9 p I 112 420 488 336 194 90 34 8

i 2 3 4 5 6 7 8 9 1 0 p , 114 456 536 422 216 106 56 14 2

i 2 3 4 5 6 7 8 9 1 0 1 1 1 2 pt 100 504 692 606 396 250 132 38 12 6 2

i 2 3 4 5 6 7 8 9 1 0 1 1 p , 108 448 828 740 584 322 176 96 48 12

i 2 3 4 5 6 7 8 9 1 0 1 1 1 2 pt 86 580 858 796 552 358 232 136 60 24 6

Code C[20,722,4], r(C) = 8

Code C[24,1058,6], r ( C ) = 8

Code C[30,1682,8], r ( C ) = 9

Code C[32,1922,10], r ( C ) = 10

Code C[38,2738,12], r ( C ) = 12

Code C[42,3362,14], r(C) = 11

Code C[44,3698,14], r ( C ) = 12

Obviously, the estimates of distances and MRDS are rather inaccurate and at least for short lengths the actual parameters of the constructed codes are better. We give a short table of balanced codes g(p,2), constructed with the help of a computer (Table VI). We list also the RDS-spectra of the codes.

Observation I : Codes similar to those of this section were studied by V. Levenstein [17, Section 71 and by A. Tietavainen [24]. Both authors have obtained the distance estimate (11) and applied it to deriving lower bounds for the exponential sums. In contrast, we apply the Weil upper bound to estimate the distance of the code 4. This observation is due to V. I. Levenstein.

Observation 2: The constant in the Vinogradov-Polya inequality (7) can be improved. Write (7) in the form rp I ( y + oX1))fi In p , where y is a constant. It follows from the results of [8] that y I y1 = 4/7r2. Moreover, it follows from [8] that, for p > 10, the incomplete exponen- tial sum satisfies rp I (5/7r2)fi In p . The latter estimate, when substituted in (8), proves to be very tight for short code lengths (cf. Table VI). The paper [13] provides still better asymptotic results. For example, for p 1 mod4, Theorem 2 of [13] implies y I y2 = 0.125. Of course, these improvements can be made in all our estimates that use the inequality (6) (Lemma 1 and Statements 2, 301, 4, and 6). We owe these two references to I. E. Shparlinslii.

VI. CONCLUSION

We investigated the properties of balanced dc-con- strained codes with distance approximately equal to half of the code length. These codes might be useful in data transmission channels with a high level of noise, such as magnetic recording and optical channels. In principle, at

least for short block lengths, the codes are realizable and may be encoded and decoded using a table of codewords. The mathematical framework underlying the code con- struction is the theory of (incomplete) exponential sums. Essentially, we considered a class of codes formed by values of Legendre symbols of polynomials of bounded degree on the set of residues modulo a prime. In particu- lar, taking linear polynomials, we obtained the Hadamard codes. Applying well-known estimates of the exponential sums, we computed the code parameters and proved that the proposed codes are in fact dc-constrained.

ACKNOWLEDGMENT

The authors are grateful to the participants of the coding theory seminar of the Institute for Problems of the Information Transmission under the direction of L. A. Bassalygo for a number of valuable comments. We thank also M. V. Burnashev for informing us of [231. The anonymous referees provided a number of useful sugges- tions that allowed the authors to improve the presenta- tion.

APPENDIX

Proof of Lemma 1: Let e p ( x ) = exp{27rix/p}. Define the discrete Fourier transform

Then of course

We have

b - 1 . c ep( - u z ) . (AI)

u = o

For the modulus of the first term on the right-hand side of (Al), we have

by the Weil inequality. Consider the second term:

1111 /Ill

BARG AND LITSYN: DC-CONSTRAINED CODES FROM HADAMARD MATRICES

x ( f ( x ) ) e , ( x z )

807

=fi.

REFERENCES

A. Barg and S. Litsyn, “A construction of balanced error-correcting codes,” in 2nd Int. Workshop on Algebraic and Combin. Coding Theory, Abstracts of papers, Leningrad, U.S.S.R., 1990, pp. 16-19. P. T. Bateman, S . Chowla, and P. Erdos, “Remarks on the size of L(l,x),’’ Ml. Marh. Debrecen, vol. 1, pp. 165-182, 1950. E. E. B e r g ” , k M. Odlyzko, and S . Sangani, “Half weight block codes for optical communication,” AT&T Tech. J., vol. 65,

E. R. Berlekamp, Algebraic Coding Theory. New York, McGraw- Hill, 1968. M. Blaum and H. C. A. van Tilborg, “On error-correcting balanced codes,” IEEE Trans. Inform. Theory, vol. 35, no. 5, pp. 1091-1093, Sept. 1989. A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, “A new table of constant weight codes,” IEEE Trans. Inform. Theory, vol. 36, no. 6, pp. 1334-1380, Nov. 1990. A. R. Calderbank, M. A. Herro, and V. Telang, “A multilevel approach to the design of dc-free codes,” IEEE Trans. Inform. Theory, vol. 35, no. 3, pp. 579-583, May 1989. T. Cochrane, “On a trigonometric inequality of Vinogradov,” J . Number Theory, vol. 27, no. 1, pp. 9-16, 1987.

pp. 85-93, 1986.

1101 -, “DC-constrained error-correcting codes with small running sum,” in 2nd Int. Workshop on Algebraic and Combin. Coding Theory, Abstracts of papers, Leningrad, U.S.S.R., 1990, pp. 49-51.

1111 R. H. Deng and M. A. Herro, “DC-free coset codes,” IEEE Trans. Inform. Theory, vol. 34, no. 4, pp. 786-792, July 1988.

1121 H. C. Ferreira, “Lower bounds on the minimum Hamming distance achievable with runlength constrained or dc-free block codes and the synthesis of a [16,8]d,, = 4 dc-free block code,” IEEE Trans. Magn., vol. MAG-20, pp. 881-883, Sept:1984.

[131 A. Hildebrandt, “Large values of character sums,’’ J . Number Theory, vol. 29, no. 3, pp. 271-296, 1988.

1141 K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. Berlin: Springer, 1982.

[15] J. Justesen, “Information rates and power spectra of digital codes,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 457-472, 1982.

[16] V. Kolesnik and V. Krachkovsky, “Generating functions and lower bounds on rates of limited error-correcting codes.” IEEE Trans.

(M)

(A4)

Inform. Theory, vol. 37, no. 3, pt. 11, pp. 778-788, May 1991. [171 V. I. Levenstein, ‘‘Bounds for packings of metric spaces and certain

applications,” in Prob. Kybernetiki (Prob. Cybern.), vol. 40, pp. 47-110, 1983, in Russian.

[18] R. Lid1 and H. Niederreiter, Finite Fields. London: Addison-Wes- ley, 1983.

[19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Cor- recting Codes. Amsterdam: North Holland, 1977.

[20] H. L. Montgomery and R. C. Vaughan, “Exponential sums with multiplicative characters,” Inuent. Math., vol. 43, pp. 69-82, 1977.

[21] G. Pierobon, “Codes for zero spectral density at zero frequency,” IEEE Trans. Inform. Theory, vol. IT-30, no. 2, pt. 11, pp. 435-439, Mar. 1984.

[22] W. M. Schmidt, Equatwns over Finite Fielh: An Elementary Approach, Lecture Notes Mathematics, vol. 536. Berlin: Springer, 1976.

[231 F. Spitzer, Principles of Random Walk. Princeton, N.J.: Van Nos- trand, 1964.

[241 A. Tietavainen, “Lower bounds for the maximum moduli of certain character sums,” J. Lon. Math. Soc., vol. 29, pp. 204-210, 1984.

[25] -, Incomplete Sums and Two Applicatwns of Deligne’s Result, in Lecture Notes Mathematics, vol. 1352, L. L. Avramoc and K. B. Tchakerian, Eds. Berlin: Springer, 1988, pp. 190-205.

[91 G. Cohen and S. Litsyn, “DC-constrained error-correcting codes with small running digital sum,” IEEE Trans. Inform. Theory, vol. 37, no. 3, pt. 11, pp. 949-955, May 1991.

[26] V. A. Zinoviev, “GeneralEed -concatenated codes,” Probl. Per- adach. Inform., vol. 12, no. 1, pp. 5-15, Jan./Mar. 1976. (Probl. Inform. Trans., vol. 12, no. 1, 1976.)

lossless of finite order can be viewed as “deterministic with bounded delay.” Fig. 2. Example for bound of Theorem 5.

C . . ” C. ._