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2.8 2.9 3.0 3.1 3.2 R

Fig. 7. Probability density function of the envelope R in the neighborhood of its right-hand peak for A = 2, B = 1, Solid curves represent the result of (33), dashed curves the approximation (35). Curves are indexed with the mean square value N of the noise.

[ l , (3.10)-(3.16), (3.19)]. Thus Esposito and Wilson’s [2 ] and Rice’s [3] approximations drop off to zero slightly more rapidly than the Rayleigh-Rice distribution for the sum of a sinusoid of ampli- tude Ro and narrowband Gaussian noise whose phasor components have variance N.

ACKNOWLEDGMENT

The author thanks Dr. R. Price and Dr. J. Mullen for their most useful comments and suggestions about this work.

REFERENCES S. 0. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J . , vol. 23, pp. 282-332, July 1944; vol. 24, pp. 46-156, Jan. 1945. Reprinted in N. Wax, Selected Papers on Noise and Stochas- tic Processes. R. Esposito and L. R. Wilson, “Statistical properties of two sine waves in Gaussian noise,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 176-183, Mar. 1973. S . 0. Rice, “Probability distributions for noise plus several sine waves-the problem of computation,” IEEE Trans. Commun., vol. COM-22, pp. 851-853, June, 1974. R. Price, “An orthonormal Laguerre expansion yielding Rice’s enve- lope density function for two sine waves in noise,” IEEE Trans. Inform. Theory, vol. 34, pp. 1375-1382, Nov. 1988. J . I . Marcum, “A Statistical Theory of Target Detection by Pulsed Radar,” Rand Corp. Rep. RM-753, July 1, 1948. Reprinted in IRE Trans. Inform. Theory, vol. IT-6, pp. 59-267, Apr. 1960. C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. Elmsford, N.Y.: Pergamon Press, 1968. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. W. F. McGee, “Another recursive method of computing the Q-func- tion,” IEEE Trans. Inform. Theory, vol. IT-16, pp. 500-501, July 1970. G. M. Dillard, “Recursive computation of the generalized Q-func- tion,” IEEE Trans. Aerospace Electron. Syst., vol. AES-9, pp.

C. W. Helstrom, “Computing the generalized Marcum Q-function,” to appear in IEEE Trans. Inform. Theory. SLATEC Mathematical Programming Library, Document NESC #820, National Energy Software Center, 9700 Cass Avenue, Ar- gonne, IL, 60439, n.d.

New York: Dover, 1954, pp. 133-294.

Cambridge: Cambridge Univ. Press, 1986.

614-615, July 1973.

H. B. Dwight, Tables of Integrals and Other Mathematical Data. New York: Macmillan, 1947. 1. S . Gradshteyn and I . W. Ryzhik, Tables of Integrals, Series, and Products. D. Middleton, An Introduction to Statistical Communication The- ory. New York: McGraw-Hill, 1960. S . 0. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J . , vol. 27, pp. 109-157, Jan. 1948. M. Abramowitz and I . A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Washington, DC: U.S. Government Printing Office, 1970; and New York: John Wiley, 1972.

New York: Academic Press, 1965.

A Bound for Divisible Codes

Harold N. Ward

Abstracr-A divisible code is a linear code whose word weights have a common divisor larger than one. If the divisor is a power of the field characteristic, there is a simple hound on the dimension of the code in terms of its weight range. When this bound is applied to the subcode of words with weight divisible by four in a type 1 binary self-dual code, it yields an asymptotic improvement of the Conway-Sloane bound for self-dual codes.

Index Terms-Divisible codes, self-dual codes, bounds.

I. INTRODUCTION

A divisible code is a linear code whose word weights have a common divisor greater than one. The simplest such code is a replicated code, created by repeating each coordinate of a selected code a certain number of times. If the divisor A of a divisible code is prime to the alphabet field characteristic, and no coordinate is identically zero, the code is equivalent to a A-fold replicated code [ 101. Generalized Reed-Muller codes and self-dual codes covered by the Gleason-Pierce theorem, however, provide examples of divisible codes whose divisors are powers of the field characteristic. Their dimensions are usually larger than those for replicated codes with the same divisor and length. The Gleason-Pierce theorem can be construed as describing the divisible codes encountered at the maximum conceivable dimension for the length [IO]. Satisfactory bounds on divisible codes in terms of the length and divisor are not known.

On the other hand, there is a simple bound for the dimension of a divisible code involving the weight spectrum, the list of weights codewords may have. We present this bound in Section IV, using preliminaries on characters from Section 111. As a warm-up, we employ character theory to prove the theorem of Bonisoli [l], classifying constant weight linear codes, in Section 11. Then, in Section V, we improve the following recent bound of Conway and Sloane for binary self-dual codes [3]: the minimum distance of such a code of length n is at most 2[(n + 6)/10], except for some low values of n. Although our bound is asymptotically stronger, their methods produce the detailed analysis of weight enumerators needed for classifying codes and proving existence from mass formulas.

Let F = GF(q) be a finite field of q elements, where q is a power p g of the prime p . The trace of an element a of F is the

Manuscript received January 24, 199 1. The author is with the Department of Mathematics, Math-Astronomy

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sum of its conjugates and lies in GF( p ) [4, p. 1161:

Let < be a complex primitive p th root of unity, and for legibility set

exp ( a ) = < l r ( a ) ,

where a E F. If V is a vector space over F , the characters of the additive group of V are the functions exp (A), with X in the dual space V * of V : exp ( X ) ( u ) = exp (%U)). A linear code C is a finite-dimensional space over F , regarded as a subspace of the ambient space F" of words of length n over F by means of a set X I , . . . , A, of coordinate or coding functionals. The embedding is

for c E C, it being assumed one-to-one. The weight function on C is then given by the character formula

Throughout this correspondence, codes will be assumed linear and the notation just introduced will be maintained. A standard source for background and references is the book by MacWilliams and Sloane [4].

11. CONSTANT WEIGHT CODES

The prototypical constant weight code, in which all nonzero words have the same weight, is a Hamming dual: take a space C of dimension m over F and, for the coding functionals, select one nonzero functional from each one-dimensional subspace of C* (C* is the vector space dual, not the dual code). The length of the code is ( q m - l ) / (q - 1) and the common weight q m - ' . (These are the dual codes of the classic Hamming codes.)

Theorem [l]: Let C be a constant weight code, no coordinate being identically 0. Then C is equivalent to a replicated Hamming dual.

Proof: If w is the weight function of C and A is the weight of the nonzero words, then 1 - A - ' w is the characteristic function $ of 0: $(O) = 1 and $(c) = 0, for c # 0. But

$ = q p k e x p ( h ) , k C *

where k = dim C, the dimension of C. Thus, if A,, * e , An are the coordinate functionals,

n + A - ' q-l 1 exp(aXi) .

i = l a # O

Invoking the independence of the characters of C, we find that

from the trivial character, and

A / q k - ' = numberofpa i rs (a , i ) with ahi= h,

for each X # 0. That is, each one-dimensional subspace of C* is represented by A / q k - ' of the hi. Scaling the X i in such a repre- senting set to be equal produces a code equivalent to C that is a replicated Hamming dual. The equation from the trivial character

0 corroborates the expansion of the lengths.

111. CHARACTER RESULTS

Let Q( <) be the cyclotomic field of pth roots of unity; the text by Weiss [ l l ] is a good reference for the arithmetic in (Q(<). The element 1 - < is a prime divisor of p in the ring S[<] of algebraic integers in Q ( { ) . If U is the exponential p-adic valuation, normal- ized by the demand that u(l - <) = ( p - l)- ' , then u ( p ) = 1. Thus when U is restricted to b, it becomes the usual p-adic valuation: u(n) is the exponent of the highest power of p dividing n , and u(0) = 03. (Occasionally we shall write up to emphasize the prime, and valuations of binomial coefficients will be written with- out the outside parentheses.)

Any function f on a code C over F with values in Q(<) is a unique linear combination of characters of C with coefficients in Q(<), a fact exploited in Section 11. If u ( f ( c ) ) 2 0 for all c in C , f can be read modulo 1 - < to produce a function f with values in F. The valuations of the denominators of the coefficients lead to bounds on the degree of f [9]. Here, we shall deal directly with the functions f , using the degree idea as motivation.

The equation U( p ) = 1 follows from the fact that

(1 - < ) P - ' =p.(<),

where U( () is a unit in 2[ <] that will be viewed as a polynomial in <. If k > 0 and we write k = Q ( k ) ( p - 1) + r , with 0 < r 5 p - 1, then

and

The transitivity of the Galois group implies this holds with <' in place of {, for 0 < t I p - 1; since r > 0, it also holds for t = 0. As the values of a character x of C are powers of <,

We set Q(0) = 0 (instead of - 1).

Lemma:

v ( k ! ) I Q ( k )

Proof: By the well-known formula [ 5 , p. 471,

" k k

n = I

If k > 0, the inequality is strict and the integer v ( k ! ) is at most 0 [ ( k - l ) / (P - 111.

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Let f: C + Q( <) be a function. Write f as a linear combination E a , x of characters, ax E Q({), and define

u ( f ) = minu(a,) X

Then

Proposition I : Let h ( x ) be a polynomial of degree m with integral coefficients, and suppose u(a) 2 - D , where D 2 1 ( D for denominator) and a E IQ(<). Then for any character x,

u(h (a (1 - x))) L -mD + Q ( m )

Proof: By the equation before the lemma, u((1 - x ) ~ ) 1 Q(k) . If h ( x ) = a,xm + * . . +a,, then

u ( a k ( a ( l - x))~) 2 k u ( a ) + Q ( k ) 2 -kD + Q ( k )

Since D 2 1 , this last expression is nonincreasing in k , so that it is at least -md + Q(m). As this bounds each term in h(a(1 - x)), it bounds the sum. 0

Proposition 2: Let f: C + mQ( {) be a function with f ( 0 ) E U. Let D L 1 and suppose U( f) 2 - D. Then

U( L) P -mD.

Proof: By the Vandermonde formula,

Since u ( f - f ( 0 ) ) 2 - D also, the truth of the proposition for f - f ( 0 ) will imply it for f. Thus we may assume f ( 0 ) = 0. Then f = - C a J l - x) and we may induct on the number of terms in the sum: set f = a(1 - x) +fl, where f l involves one fewer term, fl(0) = 0, u(a) z - D , and u ( f , ) 2 - D. Again we write

By Proposition 1 and the lemma,

Then by induction, the valuation of the kth term i s at least - kD - ( m - k ) D = - mD, which is the desired bound for the sum. 0

IV. DIVISIBLE CODES WITH AN INTERVAL SPECTRUM

Let C be a divisible code over F , with divisor p'. Suppose the weights of the nonzero words lie in an interval of b - a multiples of p e , from ( a + 1)p' to bpe. If w is the weight function on C,

then the function

is again the characteristic function of 0. As in Section 11,

summed over the characters of C. This equation shows that U ( $ ) =

- g dim C ( q = p g ) . From the introduction, on the other hand, q is a denominator for w , so that u(b - p - ' w ) 2 - ( g + e) . Then by the first equation for and Proposition 2,

Comparing these two results on U($), we obtain the following bound.

Divisible Code Bound: Let C be a divisible code over GF(q) , q = pg, with divisor pe. Let the nonzero weights of codewords of C be of the form be, a < 1s b. Then

g dim C 5 ( b - 0) ( g + e ) + u p ( : ) . For example, the even subcode of the binary Golay code is

divisible by 4, with nonzero weights 8 , 12, and 16. For this code g = 1 , e = 2, a = 1 , and b = 4. Thus,

and of course the bound is met. Similarly, the dual of the ternary Golay code is divisible by 3,

with nonzero weights 6 and 9. The bound gives 5, the dimension of the code. The hexacode also meets the bound, this time with g = 2, p = 2, e = 1, and weights 4 and 6.

It should be pointed out that U ( b ) can be determined without

knowing ( b ) numerically: it is the number of carries when the addition b = a + ( b - a) is done base p [6].

p a

V. SELF-DUAL CODES

As the previous examples suggest, the formally self-dual codes appearing in the Gleason-Pierce theorem are natural candidates for an application of the divisible code bound. The four nontrivial types, with their field size q, divisor A , and Mallows-Sloane bound on the minimum weight d as a function of the length n , are these [71:

n

8 n 24

I) q = A = 2, d 5 2 [ - - ] + 2 ,

11) q = 2, A = 4, d 5 4[ -1 + 4,

111) q = A = 3 , d 5 3[-] + 3 , 4 I n;

IV) q = 4, A = 2, d 5 2[-] + 2,

A great deal is known about the existence and structure of codes meeting these bounds [2]. In particular, such codes cannot exist for arbitrarily large n.

For the binary types one applies the divisible code bound to

n even;

8 I n;

l 2 n 6

n even

once-shortened codes to avoid the high value of b from the all-1 word; one can try shortened codes for the other types, too. None of these attempts improve the bounds above, although they come close. For example, when n = 24m in a type 11) code, the suggested minimum weight leads to the demand

2 I v2( 5“,, 1 )

But in the addition 5m - 1 = m + (4m - 1) done base 2, there are always at least two carries; so this inequality holds for any m.

Suppose, however, that C is a type I) code that is genuinely self-dual. On C , we take the familiar formula

w ( x + y ) = W ( X ) + W ( Y ) - 2 w ( x y ) ,

xy being the component-wise product. Divide it by 2 and read the result modulo 2, letting h(x) = w(x)/2 (mod 2). Then, because w(xy) = x . y (mod 2), we find

for x, y in C , since x . y = 0. That is, X is a linear functional on C with values in GF(2). Its kernel CO is the set of words with weight divisible by 4, and we can apply the divisible code bound to it. This well-known code CO was used in [8] to rule out certain extrema1 codes, and it figures in the work of Conway and Sloane [3] cited in the introduction.

Theorem: The minimum weight d of a binary self-dual code of length n satisfies

n 2

6 3 d i - + 2 + -log, n

Proof: Write n = 8m + 2 r , 0 5 r 5 3. If the code is of type 11), r = 0 and

2

Otherwise, the proofs are all minor variations of this one for r = 1: the code C, has dimension 4m. Let 4a + 2 5 d I 4 a + 4. If d = 4 a + 4, the highest weight in CO is at most 8m - 4 a - 4 because of the all-1 word. Thus, in the divisible code bound, b = 2 m - a - 1, and

2m-:- l ) . 4 m 5 3(2m - a - 1 - a) + U(

Then,

n 5 2

6 3 3 d = 4 ~ + 4 5 - + - + - U

On the other hand, if d = 4 a + 2, then b = 2m - a, and one finds

The number of carries in x = ( x - y ) + y done base 2 is at most

log, x, so that U($) I log, x. In the previous binomial coefficients x < n/4.

As the bound stands, it does not match or improve on the Conway-Sloane bound until n 2 154. In individual cases, how- ever, one should retain the binomial coefficient. For example, there are several lengths below 72 for which the largest minimum weight of a type I) self-dual code is 2 less than the bound 2[(n + 6)/10] of Conway and Sloane, as revealed by their examination of possible weight enumerators. When n = 54, 58, 64, 66, and 68, the divisi- ble code bound produces this lower value. For instance, suppose d = 14 when n = 64. In the code C, once shortened, the weights can run from 16 to 48 and the dimension is 30. Then a = 3, b = 12, and the bound would give

3(12 - 3) + v2( ’:) = 29.

Consequently, one must have d I 12.

ACKNOWLEDGMENT

The author wishes to thank N. J. A . Sloane for his advice, and the referees for their helpful comments.

REFERENCES A. Bonisoli, “Every equidistant linear code is a sequence of dual Hamming codes,” Ars Combinatoria, vol. 18, pp. 181-186, 1983. J . H. Conway and N. J . A. Sloane, Sphere Packings, Lattices and Groups. New York: Springer-Verlag, 1988. -, “A new upper bound on the minimal distance of self-dual codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1319-1333, Nov. 1990. F. J. MacWilliams and N. J . A. Sloane, The Theory of Error-Cor- recting Codes. Amsterdam: North-Holland, 1977. T. Nagell, Introduction to Number Theory. New York: Wiley, 1951. D. Singmaster, “Notes on binomial coefficients I-A generalization of Lucas’ congruence,” J. Lon. Math. Soc., vol. 8 , no. 2, pp. 545-548, 1974. N. J. A. Sloane, “Self-dual codes and lattices,” in Relations Between Combinatorics and other Parts of Mathematics, Proc. Sympos. Pure Math.. vol. 34, Amer. Math. Soc., Providence, RI, 1979, pp. 273-308. H. N. Ward, “A restriction on the weight enumerator of a self-dual code,’’ J . Combinat. Theory, Ser. A, vol. 21, pp. 253-255, 1976. -, “Combinatorial polarization,” Discrete Math., vol. 26, pp. 185- 197, 1979. -, “Divisible codes,” Arch. Math. (Basel), vol. 36, pp. 485-494, 1981. E. Weiss, Algebraic Number Theory. New York: McGraw-Hill, 1963.

Lower Bounds on t [ n , k ] from Linear Inequalities

Zhen Zhang and Chiaming Lo

Abstract-The linear inequality method for covering codes developed earlier is used to improve the lower bounds of t [ n , k ] , the smallest covering radius of any [ n , k ] binary linear code. To make better use of

Manuscript received June 22, 1990; revised January 16, 1991. This work was supported in part by the National Science Foundation under Grant NCR-8905052.

The authors are with the Communications Science Institute, Department of Electrical Engineering-Systems, University of Southem California, Los Angeles, CA 90089-0272.

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