i2fd0;:::;n?d0g. jD(i)j and the same argument as before gives the result stated below. Theorem 3 Let C be a sequence of binary linear even-weight codes ...
On Covering Radius and Discrete Chebyshev Polynomials Iiro Honkala
Department of Mathematics, University of Turku, FIN-20014 Turku, Finland
Tero Laihonen
Department of Mathematics, University of Turku, FIN-20014 Turku, Finland
Simon Litsyn
Department of Electrical Engineering-Systems, Tel-Aviv University, Ramat-Aviv, 69978, Israel
Turku Centre for Computer Science TUCS Technical Report No 81 December 1996 ISBN 951-650-924-X ISSN 1239-1891
Abstract We derive a new upper bound on the covering radius of a code as a function of its dual distance. This bound improves on the Honkala-Litsyn-Tietavainen bound and in a certain interval it is also better than Tietavainen's bound.
TUCS Research Group Coding Theory Group
1 Introduction The problem of nding bounds on the covering radius of a code as a function of its dual distance has been studied with growing interest for a few decades. In 1973 Delsarte [3] proved that the covering radius of a binary code is at most the number of non-zero weights in the dual code and ve years later Helleseth, Kl ve and Mykkeltveit gave the so-called Norse bounds [5]. Norse bounds were generalized in [11, 12]. Some very powerful results were obtained in 1990 [16, 17] by Tietavainen, who proved the following two asymptotic bounds: 1) Let C = (Cn )1 n be a sequence of binary codes Cn of length n, dual 0 0 distance d = d (n) and covering radius R = R(n) where d0=n ! 0 and R=n ! when n ! 1. Then =1
q 21 (1 ? 0(2 ? 0)):
2) Let 0 < 0 < 1=2. There are sequences C such that
H ? (1 ? H (0)) 1
2
2
where H (x) = ?x log x ? (1 ? x) log (1 ? x) is the binary entropy function. In [2] Delorme and Sole gave a better result for binary linear even-weight codes. This result was generalized by Sole and Stokes [13]. Furthermore, in 1996 Litsyn and Tietavainen [9] introduced a new approach for binary linear codes that generalizes the method presented in [4] and [15]. Using this approach and (regular) Chebyshev polynomials they were able to improve on the previous bounds. Very recently Honkala, Litsyn and Tietavainen [6] used the so-called Elias argument for further improvements. They obtained the following result: 2
2
2
q
H ( ? 0(1 ? 0)) p0 0 log ? ??0 ? 1 2 2
1
+
2
q
+2
(1
)
1
where = ? ( ? 0): Now we show that using discrete Chebyshev polynomials instead of regular ones we get a small improvement on the previous asymptotic bound. The improvement is based on the fact that we are interested in nding a good polynomial only in the discrete points of the interval [d0; n] and not in the whole interval [d0; n]. 1 2
1 2
1 2
1
2 A uniform method for bounds on covering radius Let C be a binary code of length n, dual distance d0 and covering radius R. In this section we introduce a uniform approach to get upper bounds on covering radius. This approach is alternative to the one in [1, Chapter 8] and it generalizes the method due to Litsyn and Tietavainen (see [9]) to nonlinear (and nonbinary) codes. We begin with a very natural vector equation concerning the covering radius of a code. Assume that c 2 C and x 2 St = fy 2 IFnq jw(y) = tg where w denotes the Hamming weight. We denote the number of solutions (c; x) 2 C St to the equation c + x = b (b 2 IFnq ) (1) by N (b; t). If for all b 2 IFnq the number of solutions is positive (at least) for some t, when studying all the integers t from zero up to some xed r, then the covering radius is at most r. Character sums have shown to be a powerful tool in solving the number of solutions for equations like (1). Let us now consider brie y character sums. Let q = pr where p is the characteristic of the nite eld IFq and let u 2 IFnq. It is easy to check (see, e.g., [7]) that the character u of (IFnq; +) is of the form Tr uv u (v) = ! where ! denotes a primitive complex pth root of unity, u v the usual inner product of the vectors u and v, and the trace function Trpq : IFq ! IFp is de ned by Trpq(x) = x + xp + : : : + xp ?1 : For k = 0; 1; ::: the Krawtchouk polynomial Kk (x) = Kk (x; q; n) is de ned by ! ! k X n ? x x j Kk (x) = (?1) j k ? j (q ? 1)k?j ; q p(
)
r
where
j =0 !
x = x(x ? 1) (x ? j + 1) : j j! There is a very important relation between characters and Krawtchouk polynomials (see, e.g., [7, p. 42]): Let u 2 IFnq. Then X (2) u(y) = Kk (w(u)): y2Sk
2
Let us now return to the solutions of the equation (1). We shall make use of the well known result (see [7, p. 5]) X n u(a) = q a;0 u2IFnq
where i;j is the Kronecker symbol. Therefore, we obtain by (2) X X qnN (b; t) =
X
c2C x2St u2IFnq
=
n X X
u
i=0 w(u)=i
u
(c + x ? b)
(?b)
X c2C
u
(c)Kt(i):
(3)
We de ne the weight distribution of the translate ?x + C (x 2 IFnq) to be the (n + 1)-tuple A(x) = (A (x); : : :; An(x)) where Ai(x) = jfc 0 2 C j w(? x + c) =0 igj: Since we have for the MacWilliams transform A (x) = 0 (A (x); :::; An(x)) of A(x) that (see, e.g., [17, p.35]) X X A0i(x) = jC1 j u(?x) u (c); u2S c2C 0
0
i
the equation (3) implies
n 0 X Ai( )Kt(i): i=0 2 IFnq we have Pnt=0 tN (
b
qnN (b; t) = jC j
It is clear that if for all b b; t) > 0 where t 0 for t = r + 1; :::; n, then the covering radius R is at most r. Combining this with the previous formula we get the following. Theorem 1 Let C be a code and let f (x) = Pnin iKi(x) where i 0 for all i = r + 1; :::; n. Assume that for each b 2 IFq =0
f (0) +
n 0 X Ai ( i=1
b)f (i) > 0:
(4)
Then the covering radius R of the code C is at most r.
Proof. Since A00(x) = 1 for all x 2 IFnq, we have for all b 2 IFnq n 0 X 0 < f (0) + Ai( i=1
n X n ? k tN (b; t): b) tKt (i) = q t=0 t=0
n 0 X )f (i) = Ai( i=0
b
n X
Consequently, N (b; t) 6= 0 for some t r and so the claim follows. 3
2
3 Discrete Chebyshev polynomials in covering radius problem In order to use Theorem 1 eciently we should nd a 0polynomial such that jf (i)j is small compared to f (0) whenever i 6= 0 and Ai(b) 6= 0. Let ! ! ! r X m ? x x r j m Dr (x) = Dr (x) = (?1) j j r ? j j be the discrete Chebyshev polynomial of degree r on the interval [0; m] (see [14, p.33]). These polynomials satisfy [14, p.34] ! ! m X 2 r m + r + 1 (5) Dr (i)Dl(i) = r;l 2r + 1 r : i It is known [14, p.39,p.28] that q(x) = Drm (x)=Drm (0) gives the minimum Pm value of i p(i) among the polynomials p(x) of degree at most r with p(0) = 1. Let C = (Cn)1 n be a sequence of binary linear codes of length n, dimension k = k(n), dual distance d0 = d0(n) and covering radius R = R(n) where R=n ! and d0=n ! 0 when n ! 1. Assume also that 0 < 0 < 1=2: Let be a positive number less than the Elias range, i.e., p < := 21 (1 ? 1 ? 20) and denote l = bnc: We choose now in (4) m 0 f (x) = D(x) := DDr m(x(??dd0) ) r 0 where m = n ? l ? d . Notice that the interval [d0; n ? l] is now moved to [0; m]. Notice further that r r! d0 + j ? 1! m + d0 ! X 0 (6) Dr (?d ) = r ?j : j j j We shall need the following lemma. Lemma 1 With the assumptions above we have X 0 AijD(i)j ! 0 =0
=0
=1
2
=1
=0
in?l
where n ! 1 and r ! 1 and A0i denotes the number of words of weight i in C ? .
4
Proof. As in the proof of the Elias bound [8, p.61], we have
0 Ai ? + 0 ; in?l X
1 2
0
1 ? maxi2fD; (;:::m r ?d0 ) i 01
=1
and therefore, by (5), the condition 0 2n?k rDr (?d) +1
m+r+1 2r +1
(8)
r r
2
implies by Theorem 1 that R r. Obviously any summand of the sum (6) provides a lower bound on D(?d). We take %
$
p j = 2(r +r m) (r ? 2d0 + 4d0 + r + 4d0 m) : Consequently, we obtain by (8) that R r, if 2
!
2
!
!
!
!
0 + j ? 1 m + d0 m + r + 1 ? = 2r ? = d r : (9) j r 2r + 1 r?j j Let limn!1 j=n = and limn!1 r=n = . We recall a well known result (see, e.g., [10, p. 310])
2n?k+1
!
2aH2 a 2aH2 q b 8a (1 ? ) ( )
( )
where = b=a. 5
1 2
1 2
(10)
Combining (9) and (10) with the McEliece-Rodemich-Rumsey-Welch bound n ? k < H ( 1 ? q0(1 ? 0)); n 2 we obtain the asymptotic upper bound on covering radius from the least positive number satisfying the inequality ! ! q 1 0 H ( 2 ? 0(1 ? 0)) H + ( + )H 0 + (11) ! 1 2 ? 0 +(1 ? )H 1 ? ? 2 ( + 1 ? ? )H + 1 ? ? 0 ? : Since this is true for all < we get the following result. Theorem 2 Let be the smallest positive number satisfying the inequality (11) where is replaced by . Then we have for the sequence C . This bound gives a small improvement on the Honkala-Litsyn-Tietavainen bound (for instance at 0 = 0:28 the improvement is 0.0016) and it is better than Tietavainen's bound when 0 0:278. For binary linear even-weight codes we have A0i(b) = 0 when i 2 (0; d0) [ (n ? d00; n). Now we take in (4) f (x) = D(x) where m = n ? 2d0. Noticing that An(b) = (?1)w b we can choose the parity of r in such a way that 0 An(b)D(n) = 1. Therefore 2
2
2
2
2
2
( )
n 0 X D(0) + Ai( i=1
b)D(i) > 2 ? 2n?k i2fdmax 0;:::;n?d0 g jD(i)j
and the same argument as before gives the result stated below. Theorem 3 Let C be a sequence of binary linear even-weight codes satisfying the condition of Theorem 2. Let be the smallest positive number satisfying the inequality ! ! q 1 0 H ( 2 ? 0(1 ? 0)) H + ( + )H 0 + ! 1 2 ? 0 0 +(1 ? )H 1 ? 0 ? 2 ( + 1 ? 2 )H + 1 ? 20 ? : Then . The previous bound improves on Tietavainen's bound when 0 0:263. It also improves on the bound due to Litsyn and Tietavainen for even-weight codes (see [9, Theorem 3]). For example, at 0 = 0:27 the improvement is 0.0022. Acknowledgements. We would like to thank I. Krasikov for inspiring discussions. 2
2
2
2
2
6
References [1] Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering Codes. Elsevier, in preparation [2] Delorme, C., Sole, P.: Diameter, covering index, covering radius and eigenvalues. Europ. J. Combin. 12, 95{108 (1991) [3] Delsarte, P.: Four fundamental parametres of a code and their combinatorial signi cance. Information and Control 23, 407{438 (1973) [4] Helleseth, T.: On the covering radius of cyclic linear codes and aritmetic codes. Discrete Appl. Math. 11, 157{173 (1985) [5] Helleseth, T., Kl ve, T., Mykkeltveit, J.: On the covering radius of binary codes. IEEE Trans. Inform. Theory 24, 627{628 (1978) [6] Honkala, I., Litsyn, S., Tietavainen, A.: On algebraic methods in covering radius problems. In: Cohen, G., Giusti M., Mora, T. (eds.) Applied Algebra, Algebraic Algoritms and Error-Correcting Codes. Lecture Notes in Computer Science, Vol 948. Berlin, Heidelberg, New York: Springer 1995 [7] Honkala, I., Tietavainen, A.: Codes and number theory. In: R. A. Brualdi, W.C. Hu man, and V.S. Pless (eds.) Handbook of Coding Theory, to appear. [8] van Lint, J.: Introduction to Coding Theory. Berlin, Heidelberg, New York: Springer-Verlag 1982 [9] Litsyn, S., Tietavainen, A.: Upper bounds on the covering radius of a code with a given dual distance. Europ. J. Combin., 173, 265{270 (1996) [10] MacWilliams, F., Sloane, N.: The Theory of Error-Correcting Codes. Amsterdam: North-Holland 1977 [11] Sole, P.: Asymptotic bounds on the covering radius of binary codes. IEEE Trans. Inform. Theory 36, 1470{1472 (1990) [12] Sole, P., Mehrotra, K.: Generalization of the Norse bounds to codes of higher strength. IEEE Trans. Inform. Theory 37, 190{192 (1991) [13] Sole, P., Stokes, P.: Covering radius, codimension, and dual-distance width. IEEE Trans. Inform. Theory 39, 1195{1203 (1993) 7
[14] Szego, G.: Orthogonal Polynomials. Colloquium Publications, vol. 23, New York: American Math. Soc., 1959 [15] Tietavainen, A.: Codes and character sums. Springer Lecture Notes in Computer Science 388, 3{12 (1989) [16] Tietavainen, A.: An upper bound on the covering radius as a function of its dual distance. IEEE Trans. Inform. Theory 36, 1472{1474 (1990) [17] Tietavainen, A.: Covering radius and dual distance. Designs, Codes and Cryptography 1, 31{46 (1991)
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