On Upper Bounds for Minimum Distance and ... - Semantic Scholar
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On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes TERO LAIHONEN
[email protected]
Department of Mathematics, University of Turku, FIN-20500 Turku, Finland
SIMON LITSYN
Department of EE-systems, Tel-Aviv University, Ramat-Aviv 69978 Israel
[email protected]
Abstract. We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietavainen [10] and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The new upper bound on the information rate is an application of a shortening method of a code. These results improve on the best presently known asymptotic upper bounds on minimum distance and covering radius of non-binary codes in certain intervals. Keywords: Minimum distance, covering radius, asymptotic information rate, dual distance.
1. Introduction Bounds on minimum distance and covering radius have attracted a great deal of research (see, e.g. [12], [4]). In this paper we consider the case of bounds for parameters of non-binary codes. The best presently known upper bound on the minimum distance is due to M. Aaltonen [2], and was obtained using the linear programming method in the generalized Johnson scheme (see Section 3 for more details). It is known that the covering radius depends crucially on the dual distance. In 1973 Delsarte [6] proved that the covering radius of a code is at most the number of nonzero weights in the dual code. Later in the papers [5], [7], [8], [16], [17], [18], [20], [21] a number of bounds have been obtained for the covering radius of a code with a given dual distance. Especially, Tietavainen [21] gave the following asymptotic result: n 0 0 Let (Cn )1 n=1 be a sequence of codes Cn Fq with dual distance d = d (n) and 0 0 covering radius R = R(n) where R=n ! and d =n ! when n ! 1. Then q ? 1 q ? 2 0 1p (1) ? 2q ? q (q ? 1)0 (2 ? 0 ): q In the paper [18] Sole and Stokes proved the following asymptotic result for linear codes with certain assumptions (see [18], Section VI):
Hq
q? q
1
p
? q?q 0 ? q (q ? 1)0 (1 ? 0 ) 2
logq
2
? ? 0 ?1
q 1 (1 )q
(2)
2
where Hq (x) =
0
if x = 0; x logq (q ? 1) ? x logq x ? (1 ? x) logq (1 ? x) if 0 < x q?q 1 :
In the last expression an upper bound on the information rate is implicitly used. The best known bound for non-binary codes was obtained by M.Aaltonen [2] in the frames of linear programming method. In this paper we generalize a method due to Litsyn and Tietavainen [10] to nonbinary codes and we give a new upper bound on the asymptotic information rate improving on the Aaltonen's bound; i.e., a new asymptotic upper bound on the minimum distance is obtained. Combining these two results gives a new asymptotic upper bound on the covering radius of non-binary linear codes which improves on the best presently known bounds (1) and (2) in certain intervals (see Section 4).
2. The generalized method Let Fq denote the nite eld of cardinality of q. Assume that C Fqn is a linear code of dimension k, minimum distance d( 3), covering radius R and dual distance d0 . Let the (n ? k) n matrix H = (h1 ; : : : ; hn ) be a parity check matrix for C and, denote the set fh1 ; : : : ; hn g by L and the nonzero elements of Fq by Fq? . Let Na (L; s; b), where a = (a1 ; : : : ; as ) 2 (Fq? )s , be the number of solutions (x1 ; : : : ; xs ) 2 Ls of the equation a1 x1 + : : : + as xs = b:
Denote also N (L; s; b) =
P
a2(F?q )s
(3) Na (L; s; b).
The covering radius R of a linear code C is the smallest integer r such that every syndrome of C is a Fq -linear combination of at most r columns of H . Let q = pr where p is the characteristic of Fq . We recall (see e.g. [6]) that a character u , u 2 Fqn , of (Fqn ; +) is of the form u (v) = ! T rp (uv) q
for all v 2 Fqn
where ! denotes a primitive complex pth root of unity, u v the inner product of the vectors u and v, and the trace function T rpq : Fq ! Fp is de ned by T rpq (x) = x + xp + : : : + xp
r?1
:
The next lemma is crucial in the sequel and it generalizes the result presented in [10] to non-binary codes. Lemma 1 Assume that for each b 2 Fqn?k there is a polynomial of degree at most r such that
3
f (0) +
n X i=1
where i (b) =
i (b)f (i) > 0
P
k (?b).
k2Fqn?k ;w(kH )=i
Then R r.
Proof: It is well-known (see e.g. [12], p.143) that qn?k if a = 0, X k (a) = 0
k2Fqn?k
otherwise,
and therefore, by (3), we obtain qn?k N (L; s; b) =
X k2Fqn?k
0 XX k (?b) @
x2L a2Fq?
1s k (ax)A
Furthermore,
XX
x2L a2Fq?
k (ax) = n(q ? 1) ? qw(kH )
where w denotes the Hamming weight. Since kH runs through all elements of the dual code C ? of C , when k runs through the elements of Fqn?k , we have qn?k N (L; s; b) =
and therefore, qn?k N (L; s; b) =
0 n X @ i=0
n X i=0
X k2Fnq?k ;w(kH)=i
1 s k (?b)A (n(q ? 1) ? qi)
i (b)(n(q ? 1) ? qi)s :
(4)
Pr
We choose next such a polynomial g(x) = s xs that g(n(q ? 1) ? qi) = f (i). s=0 Since 0 (b) = 1 for all b 2 Fqn?k , we have by (4) 0 < f (0) +
n X
r X
i=1
= qn?k
s=0
i (b)f (i)
s N (L; s; b):
Hence N (L; s; b) 6= 0 for at least one s (s = 0; 1; : : : ; r) and so R r.
(5)
4
We should now nd a polynomial of a low degree such that jf (i)j is small compared to f (0) when i 6= 0 and i (b) 6= 0. The Chebyshev polynomial of the rst kind and degree r is de ned in [14], p.5 by 1 x + px2 ? 1r + x ? px2 ? 1r : Tr (x) = 2 So clearly, x 1, p 1 Tr (x) ((x + x2 ? 1)r + 1) (6) 2 Assume that 0 a < b. Among the polynomials pr (x) of degree at most r such that pr (0) = 1 the one de ned by tr (x) =
b a? x bb?aa
Tr
+
Tr
2
+
?
b a
provides (see [19], p.42) the minimum of maxx2[a;b] jpr (x)j: Furthermore, max jt (x)j = 1 2
x [a;b]
r
Tr
b+a b a
?
In order to apply the polynomial tr (x) to Lemma 1 eciently, we need to know something about the asymptotic information rate of non-binary codes. It will be studied in the next section. As pointed out by a referee, our approach is analogous to the method developed by Chung et al. [3] (see also Lubotszky et al. [11]) for estimation of the girth of Cayley graphs.
3. Straight-line bound Let Mq (n; d) denote the number of words in the largest code C Fqn with minimum distance at least d. We de ne the asymptotic information rate Rq () (0 1) by 1 R () = lim sup log M (n; d); q
!1 n
n
q
q
where limn!1 d=n = : The tightest presently known upper bounds on asymptotic information rate of non-binary codes are the following ones (see [2], p.141): Rq () 1 ?
q
log (q ? 1); q?2 q
q > 2; 0
q ? 2
2
q
;
(7)
5
and Rq () 1 ? Hq (!) + fq (; );
(8)
where the parameters satisfy the following conditions: ? 2 !; 0 ? minf! ? ; 1 ? !g; 0 ! 1; 0 qq ? 1 = (1 ? )h
! ? ? ! ? q ? 1 ; ; 1? 1?
2 + (! ? )kq?1
!?
q?2
with the following notations: fq (; ) = Hq ( ) + Hq (= ) ? ( + ) logq (q ? 1) + logq (q ? 2); kq (x) =
2 p(q ? 1)x(1 ? x); (0 x 1); q?1 q?2 ? x? q q q
and h(x; y) =
x(1 ? x) ? y(1 ? y) p ; (0 x 1; 0 y 1): 1 + 2 y(1 ? y)
As shown by M. Aaltonen (see [2], p. 141), with a certain choice of parameters the bound (8) reduces to a simpler form, which is a (straightforward) generalization of the so-called rst McEliece-Rodemich-Rumsey-Welch bound [13]: q?1 : (9) R () H (k ()); 0 q
q
q
q
This bound is useful when is close to (q ? 1)=q (see [2], p.157). We are here interested in large values of , since the method presented in Lemma 1 improves on the bound (1) when 0 is large. In order to give a better upper bound on Rq (), we shall need the following theorem, which generalizes the well-known result (see e.g. [12], p.43) Mq (n; d) qt Mq (n ? t; d)
where t n ? d. Let Br (x) be the Hamming sphere of radius r and with center at r x 2 Fqn. Denote its cardinality by Vq (n; r) = P ?ni(q ? 1)i : i=0
6
Theorem 1 Let 0 d n, d ? 2r n ? t, 0 r t and 0 r 21 d: Then
Mq (n; d)
qt M (n ? t; d ? 2r): Vq (t; r) q
(10)
Proof: Let a code C Fqn be such that its cardinality is Mq (n; d) =: M ; i.e., let
C be an (n; M; d) code. We shorten the code C choosing t components of codewords and taking those codewords in which the chosen t components belong to a Hamming sphere of radius r. Finally, we delete these t coordinates. Next we show that in this way we get from C an (n ? t; M q t Vq (t; r ); d ? 2r )
code. The rst parameter is clear and the third one follows from the fact that the deleted parts of the selected codewords di er at most in 2r positions. Let us now consider the second parameter. We denote the M words (not necessarily distinct) of the t components by y1 ; : : : ; yM 2 Fqt in some order. Let x (y) =
Since 1
1
M XX
qt x2F t i=1
if y 2 Br (x); 0 otherwise.
x (yi ) =
1
M X X
(y ) qt i=1 x2F t x i q = M V (t; r); qt q
q
there exists a sphere of radius r which contains at least y1 ; : : : ; yM and so the claim follows.
(11)
M qt
Vq (t; r) of the words
By the previous theorem we get now the following upper bound on the asymptotic information rate. Theorem 2 Let 0 q?q 1 , ? 2 1 ? , 0 12 and 0 q?q 1 . Denote x2 = and y = 1??2 . Assume that x 6= . Then
Rq () R(y) + (1 ? Hq (x=2) ? R(y))
?y x?y
(12)
where R(y) is an upper bound on the asymptotic information rate at point y. Proof: Let r = b ?1 b ncc and t = b nc. It is well-known (see e.g. [9], p.55) that logq Vq (n; b nc) lim = Hq ( ) n!1 n
where 0 q?q 1 . Combining this result with Theorem 1 gives
7
Rq () (1 ? Hq (= )) + (1 ? )Rq
? 2 1?
:
Thus Rq () R(y) + ((1 ? Hq (x=2) ? R(y))
?y : x?y
By the Hamming bound (see e.g. [9], p.60): H () := 1 ? Hq (=2); 0 1;
we may write the bound (12) in the form Rq () R(y) + (H (x) ? R(y))
?y x?y
with the assumptions of the previous theorem. Hence Theorem 2 means that Rq () is on or below any straight line segment between the Hamming bound and a given upper bound, i.e., a straight line between any point on the Hamming bound (x; H (x)) and any point on a given upper bound (y; R(y)) is also an upper bound on the asymptotic information rate. Clearly, the best improvements are achieved when the line (12) is tangential to the Hamming bound and to the given upper bound. Choosing the given upper bound to be the bound (9), the bound (12) gives a small improvement on the bounds (7) and (8) in a certain interval. In Table 1 the comparing of these bounds is given for q = 16 and only those values of are given where these improvements occur on best of the bounds (7) and (8). On the other hand, Table 2 shows (for some q's) the interval [a,b] in which the bound (12) improves on the bounds (7) and (8) (note that q is at least 7). In Table 2 the parameters x and y are also shown. If we choose x = 0 and R(y) to be the bound (9) and we minimize the right-hand side of the inequality (12) (for minimization see [1], p.156), we obtain the bound (7). Our bound in Theorem 2 is analogous to the straight-line bound of [15], although its derivation is purely combinatorial. Table 1. Table 2.
8
Table 1. Numerical values for q = 16.
0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
(7) & (8) 0.70201 0.68056 0.65908 0.63755 0.61598 0.59435 0.57269 0.55098 0.52923 0.50744 0.48561 0.46374 0.44184
(12) 0.70192 0.67990 0.65788 0.63586 0.61384 0.59183 0.56981 0.54779 0.52577 0.50375 0.48173 0.45972 0.43770
0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76
(7) & (8) 0.41955 0.39722 0.37490 0.35257 0.33025 0.30792 0.28560 0.26327 0.24095 0.21862 0.19630 0.17397 0.15165
(12) 0.41568 0.39366 0.37164 0.34963 0.32761 0.30559 0.28357 0.26155 0.23953 0.21752 0.19550 0.17348 0.15146
Table 2. The interval [a,b] and parameters x and y.
a b x y q a b 7 0.54 0.56 0.08 0.58 27 0.14 0.84 8 0.52 0.60 0.08 0.62 32 0.12 0.86 16 0.26 0.76 0.06 0.78 64 0.06 0.92 q
x
y
0.04 0.86 0.04 0.88 0.02 0.94
4. New upper bounds for covering radius We are now in a position to state the results for covering radius. Theorem 3 is valid in the whole interval [0; 1] whereas Theorem 4 improves it in a certain part of this interval. Theorem 3 Let (Cn )1 n=1 be a sequence of nonbinary linear codes Cn of length n, 0 and covering dual distance d radius R where R=n ! and d0 =n ! 0 when n ! 1. ?
If 0 0 q?q 2 2 , then
1 ? 0 q?q 2 logq (q ? 1) p ; logq (1+1?00 )2
(13)
? and, if q?q 2 2 0 q?q 1 ,then
Hq(kq (p0 ))
logq
0 ?0
(1+ )2 1
:
Proof: We choose f (x) = tr (x), a = d0 and b = n. Then
(14)
9
max jf (x)j =
x [d0 ;n]
2
Thus f (0) +
n X i=1
Tr
n1 d0 +
n d0
?
1 ? (qn?k ? 1) i2max jf (i)j d0 ;n
i (b)f (i)
> 1?
Tr
qn?k
n+d0 n d0
?
[
(15)
]
:
Therefore, by Lemma 1, we have R r if qn?k Tr
n + d0 n ? d0
(16)
:
Combining the results (16) and (6) with the dual forms of the bounds (7) and (9); i.e., k lim sup n ? n n!1
1 ? 0 q ?q 2 logq (q ? 1); 0 0 q ?q 2
2
;
k Hq (kq (0 )); 0 0 q ?q 1 lim sup n ? n !1
n
gives the desired result. Note that the bounds (13) and (14) coincide at ((q ? 2)=q)2 . If we replace the dual forms of the bounds (7) and (9) with the dual form of the bound (12) where we have chosen R(y) = Hq (kq (y)); i.e., k Hq (kq (y)) ? 1 + Hq (x=2) 0 ( ? x) + 1 ? Hq (x=2) lim sup n ? n y?x !1
n
(17)
where 0 x 0 y q?q 1 , x 6= y, the same argument as in the previous proof gives the following result. We denote the right-hand side of (17) by Lx;y (0 ). n Theorem 4 Let (Cn )1 n=1 be a sequence of nonbinary linear codes Cn Fq with 0 0 0 dual distance d and covering radius R where R=n ! and d =n ! when n ! 1. Assume also that x 0 y, x 6= y where x; y 2 [0; (q ? 1)=q]. Then Lx;y (p0 ) : (18) logq (1+1?00 )2
10
In Figure 1 the comparison of the asymptotical results is shown for q = 16. We have used the label (16) for the bound obtained from (15) and when it is not valid (i.e. 0 > 0:78) from the bounds of Theorem 3. Notice that the bound (16) is general whereas the Delorme-Sole-Stokes bound uses additional conditions. It should be emphasized that in the binary situation further improvements are possible (see [8]).
Acknowledgements We are very grateful to M.Aaltonen for inspiring discussions, and for providing programs for calculating asymptotical bounds on the size of non-binary codes. The authors thank heartily also I. Honkala and A. Tietavainen for their valuable suggestions. We thank the referees for providing the references [3], [11] and [15]. Figure 1.
References 1. M. Aaltonen, Bounds on the information rate of a tree code as a function on the code's feedback decoding minimum distance, Ann. Univ. Turku. Ser. A I, Vol. 181 (1981). 2. M. Aaltonen, A new upper bound on nonbinary block codes, Discrete Math., Vol. 83 (1990), pp. 139{160. 3. F. R. K. Chung, V. Faber and T. A. Manteu el, An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian, SIAM J. Discr. Math., Vol. 7, (1994), pp. 443{457. 4. G. Cohen, S. Litsyn, A. Lobstein and H. Mattson, Jr., Covering radius 1985{1994. Technical Raport 94-D-025, Ecole Nationale Superieure des Telecommunications, Paris, 1994, submitted to AAECC Journal. 5. C. Delorme and P. Sole, Diameter, covering index, covering radius and eigenvalues,Europ. J. Combin., Vol 12 (1991), pp. 95{108. 6. P. Delsarte, Four fundamental parametres of a code and their combinatorial signi cance,Information and Control, Vol. 23, (1973), pp. 407{438. 7. T. Helleseth, T. Kl ve, and J. Mykkeltveit, On the covering radius of binary codes, IEEE Trans. Inform. Theory, Vol. 24 (1978), pp. 627{628. 8. I. Honkala, S. Litsyn, and A. Tietavainen, On algebraic methods in covering radius problems, In: Cohen, G., Giusti M., Mora, T. (eds.) Applied Algebra, Albegraic Algoritms and ErrorCorrecting Codes. Lecture Notes in Computer Science, Vol 948. Berlin, Heidelberg, New York: Springer (1995). 9. J. van Lint, Introduction to Coding Theory, Berlin, Heidelberg, New York: Springer-Verlag (1982). 10. S. Litsyn and A. Tietavainen, Upper bounds on the covering radius of a code with a given dual distance, Europ. J. Combin., to appear. 11. A. Lubotszky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica, Vol. 8, (1988), pp. 261{277.
11
12. F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes, Amsterdam: North-Holland (1977). 13. R. McEliece, E. Rodemich, H. Jr. Rumsey and L. Welch, New upper bounds on the rate of a code via Delsarte-MacWilliams inequalities, IEEE Trans. Inform. Theory, Vol. 23, (1977), pp. 157{167. 14. T. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons (1974). 15. C. E. Shannon, R. G. Gallager and E. R. Berlekamp, Lower bounds to error probability for coding on discrete memoryless channels. II, Information and Control, Vol. 10, (1967), pp. 522{552. 16. P. Sole, Asymptotic bounds on the covering radius of binary codes, IEEE Trans. Inform. Theory, Vol. 36, (1990), pp. 1470{1472. 17. P. Sole and K. Mehrotra, Generalization of the Norse bounds to codes of higher strenght, IEEE Trans. Inform. Theory, Vol. 37, (1991), pp. 190{192. 18. P. Sole and P. Stokes, Covering radius, codimension, and dual-distance width,IEEE Trans. Inform. Theory, Vol. 39, (1993), pp. 1195{1203. 19. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., v,. 23, Providence, RI, (1975). 20. A. Tietavainen, An upper bound on the covering radius as a function of its dual distance, IEEE Trans. Inform. Theory, Vol. 36, (1990), pp. 1472{1474. 21. A. Tietavainen, Covering radius and dual distance, Designs, Codes and Cryptography, Vol. 1, (1991), pp. 31{46.
c Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes TERO LAIHONEN
[email protected]
Department of Mathematics, University of Turku, FIN-20500 Turku, Finland
SIMON LITSYN
Department of EE-systems, Tel-Aviv University, Ramat-Aviv 69978 Israel
[email protected]
Abstract. We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietavainen [10] and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The new upper bound on the information rate is an application of a shortening method of a code. These results improve on the best presently known asymptotic upper bounds on minimum distance and covering radius of non-binary codes in certain intervals. Keywords: Minimum distance, covering radius, asymptotic information rate, dual distance.
1. Introduction Bounds on minimum distance and covering radius have attracted a great deal of research (see, e.g. [12], [4]). In this paper we consider the case of bounds for parameters of non-binary codes. The best presently known upper bound on the minimum distance is due to M. Aaltonen [2], and was obtained using the linear programming method in the generalized Johnson scheme (see Section 3 for more details). It is known that the covering radius depends crucially on the dual distance. In 1973 Delsarte [6] proved that the covering radius of a code is at most the number of nonzero weights in the dual code. Later in the papers [5], [7], [8], [16], [17], [18], [20], [21] a number of bounds have been obtained for the covering radius of a code with a given dual distance. Especially, Tietavainen [21] gave the following asymptotic result: n 0 0 Let (Cn )1 n=1 be a sequence of codes Cn Fq with dual distance d = d (n) and 0 0 covering radius R = R(n) where R=n ! and d =n ! when n ! 1. Then q ? 1 q ? 2 0 1p (1) ? 2q ? q (q ? 1)0 (2 ? 0 ): q In the paper [18] Sole and Stokes proved the following asymptotic result for linear codes with certain assumptions (see [18], Section VI):
Hq
q? q
1
p
? q?q 0 ? q (q ? 1)0 (1 ? 0 ) 2
logq
2
? ? 0 ?1
q 1 (1 )q
(2)
2
where Hq (x) =
0
if x = 0; x logq (q ? 1) ? x logq x ? (1 ? x) logq (1 ? x) if 0 < x q?q 1 :
In the last expression an upper bound on the information rate is implicitly used. The best known bound for non-binary codes was obtained by M.Aaltonen [2] in the frames of linear programming method. In this paper we generalize a method due to Litsyn and Tietavainen [10] to nonbinary codes and we give a new upper bound on the asymptotic information rate improving on the Aaltonen's bound; i.e., a new asymptotic upper bound on the minimum distance is obtained. Combining these two results gives a new asymptotic upper bound on the covering radius of non-binary linear codes which improves on the best presently known bounds (1) and (2) in certain intervals (see Section 4).
2. The generalized method Let Fq denote the nite eld of cardinality of q. Assume that C Fqn is a linear code of dimension k, minimum distance d( 3), covering radius R and dual distance d0 . Let the (n ? k) n matrix H = (h1 ; : : : ; hn ) be a parity check matrix for C and, denote the set fh1 ; : : : ; hn g by L and the nonzero elements of Fq by Fq? . Let Na (L; s; b), where a = (a1 ; : : : ; as ) 2 (Fq? )s , be the number of solutions (x1 ; : : : ; xs ) 2 Ls of the equation a1 x1 + : : : + as xs = b:
Denote also N (L; s; b) =
P
a2(F?q )s
(3) Na (L; s; b).
The covering radius R of a linear code C is the smallest integer r such that every syndrome of C is a Fq -linear combination of at most r columns of H . Let q = pr where p is the characteristic of Fq . We recall (see e.g. [6]) that a character u , u 2 Fqn , of (Fqn ; +) is of the form u (v) = ! T rp (uv) q
for all v 2 Fqn
where ! denotes a primitive complex pth root of unity, u v the inner product of the vectors u and v, and the trace function T rpq : Fq ! Fp is de ned by T rpq (x) = x + xp + : : : + xp
r?1
:
The next lemma is crucial in the sequel and it generalizes the result presented in [10] to non-binary codes. Lemma 1 Assume that for each b 2 Fqn?k there is a polynomial of degree at most r such that
3
f (0) +
n X i=1
where i (b) =
i (b)f (i) > 0
P
k (?b).
k2Fqn?k ;w(kH )=i
Then R r.
Proof: It is well-known (see e.g. [12], p.143) that qn?k if a = 0, X k (a) = 0
k2Fqn?k
otherwise,
and therefore, by (3), we obtain qn?k N (L; s; b) =
X k2Fqn?k
0 XX k (?b) @
x2L a2Fq?
1s k (ax)A
Furthermore,
XX
x2L a2Fq?
k (ax) = n(q ? 1) ? qw(kH )
where w denotes the Hamming weight. Since kH runs through all elements of the dual code C ? of C , when k runs through the elements of Fqn?k , we have qn?k N (L; s; b) =
and therefore, qn?k N (L; s; b) =
0 n X @ i=0
n X i=0
X k2Fnq?k ;w(kH)=i
1 s k (?b)A (n(q ? 1) ? qi)
i (b)(n(q ? 1) ? qi)s :
(4)
Pr
We choose next such a polynomial g(x) = s xs that g(n(q ? 1) ? qi) = f (i). s=0 Since 0 (b) = 1 for all b 2 Fqn?k , we have by (4) 0 < f (0) +
n X
r X
i=1
= qn?k
s=0
i (b)f (i)
s N (L; s; b):
Hence N (L; s; b) 6= 0 for at least one s (s = 0; 1; : : : ; r) and so R r.
(5)
4
We should now nd a polynomial of a low degree such that jf (i)j is small compared to f (0) when i 6= 0 and i (b) 6= 0. The Chebyshev polynomial of the rst kind and degree r is de ned in [14], p.5 by 1 x + px2 ? 1r + x ? px2 ? 1r : Tr (x) = 2 So clearly, x 1, p 1 Tr (x) ((x + x2 ? 1)r + 1) (6) 2 Assume that 0 a < b. Among the polynomials pr (x) of degree at most r such that pr (0) = 1 the one de ned by tr (x) =
b a? x bb?aa
Tr
+
Tr
2
+
?
b a
provides (see [19], p.42) the minimum of maxx2[a;b] jpr (x)j: Furthermore, max jt (x)j = 1 2
x [a;b]
r
Tr
b+a b a
?
In order to apply the polynomial tr (x) to Lemma 1 eciently, we need to know something about the asymptotic information rate of non-binary codes. It will be studied in the next section. As pointed out by a referee, our approach is analogous to the method developed by Chung et al. [3] (see also Lubotszky et al. [11]) for estimation of the girth of Cayley graphs.
3. Straight-line bound Let Mq (n; d) denote the number of words in the largest code C Fqn with minimum distance at least d. We de ne the asymptotic information rate Rq () (0 1) by 1 R () = lim sup log M (n; d); q
!1 n
n
q
q
where limn!1 d=n = : The tightest presently known upper bounds on asymptotic information rate of non-binary codes are the following ones (see [2], p.141): Rq () 1 ?
q
log (q ? 1); q?2 q
q > 2; 0
q ? 2
2
q
;
(7)
5
and Rq () 1 ? Hq (!) + fq (; );
(8)
where the parameters satisfy the following conditions: ? 2 !; 0 ? minf! ? ; 1 ? !g; 0 ! 1; 0 qq ? 1 = (1 ? )h
! ? ? ! ? q ? 1 ; ; 1? 1?
2 + (! ? )kq?1
!?
q?2
with the following notations: fq (; ) = Hq ( ) + Hq (= ) ? ( + ) logq (q ? 1) + logq (q ? 2); kq (x) =
2 p(q ? 1)x(1 ? x); (0 x 1); q?1 q?2 ? x? q q q
and h(x; y) =
x(1 ? x) ? y(1 ? y) p ; (0 x 1; 0 y 1): 1 + 2 y(1 ? y)
As shown by M. Aaltonen (see [2], p. 141), with a certain choice of parameters the bound (8) reduces to a simpler form, which is a (straightforward) generalization of the so-called rst McEliece-Rodemich-Rumsey-Welch bound [13]: q?1 : (9) R () H (k ()); 0 q
q
q
q
This bound is useful when is close to (q ? 1)=q (see [2], p.157). We are here interested in large values of , since the method presented in Lemma 1 improves on the bound (1) when 0 is large. In order to give a better upper bound on Rq (), we shall need the following theorem, which generalizes the well-known result (see e.g. [12], p.43) Mq (n; d) qt Mq (n ? t; d)
where t n ? d. Let Br (x) be the Hamming sphere of radius r and with center at r x 2 Fqn. Denote its cardinality by Vq (n; r) = P ?ni(q ? 1)i : i=0
6
Theorem 1 Let 0 d n, d ? 2r n ? t, 0 r t and 0 r 21 d: Then
Mq (n; d)
qt M (n ? t; d ? 2r): Vq (t; r) q
(10)
Proof: Let a code C Fqn be such that its cardinality is Mq (n; d) =: M ; i.e., let
C be an (n; M; d) code. We shorten the code C choosing t components of codewords and taking those codewords in which the chosen t components belong to a Hamming sphere of radius r. Finally, we delete these t coordinates. Next we show that in this way we get from C an (n ? t; M q t Vq (t; r ); d ? 2r )
code. The rst parameter is clear and the third one follows from the fact that the deleted parts of the selected codewords di er at most in 2r positions. Let us now consider the second parameter. We denote the M words (not necessarily distinct) of the t components by y1 ; : : : ; yM 2 Fqt in some order. Let x (y) =
Since 1
1
M XX
qt x2F t i=1
if y 2 Br (x); 0 otherwise.
x (yi ) =
1
M X X
(y ) qt i=1 x2F t x i q = M V (t; r); qt q
q
there exists a sphere of radius r which contains at least y1 ; : : : ; yM and so the claim follows.
(11)
M qt
Vq (t; r) of the words
By the previous theorem we get now the following upper bound on the asymptotic information rate. Theorem 2 Let 0 q?q 1 , ? 2 1 ? , 0 12 and 0 q?q 1 . Denote x2 = and y = 1??2 . Assume that x 6= . Then
Rq () R(y) + (1 ? Hq (x=2) ? R(y))
?y x?y
(12)
where R(y) is an upper bound on the asymptotic information rate at point y. Proof: Let r = b ?1 b ncc and t = b nc. It is well-known (see e.g. [9], p.55) that logq Vq (n; b nc) lim = Hq ( ) n!1 n
where 0 q?q 1 . Combining this result with Theorem 1 gives
7
Rq () (1 ? Hq (= )) + (1 ? )Rq
? 2 1?
:
Thus Rq () R(y) + ((1 ? Hq (x=2) ? R(y))
?y : x?y
By the Hamming bound (see e.g. [9], p.60): H () := 1 ? Hq (=2); 0 1;
we may write the bound (12) in the form Rq () R(y) + (H (x) ? R(y))
?y x?y
with the assumptions of the previous theorem. Hence Theorem 2 means that Rq () is on or below any straight line segment between the Hamming bound and a given upper bound, i.e., a straight line between any point on the Hamming bound (x; H (x)) and any point on a given upper bound (y; R(y)) is also an upper bound on the asymptotic information rate. Clearly, the best improvements are achieved when the line (12) is tangential to the Hamming bound and to the given upper bound. Choosing the given upper bound to be the bound (9), the bound (12) gives a small improvement on the bounds (7) and (8) in a certain interval. In Table 1 the comparing of these bounds is given for q = 16 and only those values of are given where these improvements occur on best of the bounds (7) and (8). On the other hand, Table 2 shows (for some q's) the interval [a,b] in which the bound (12) improves on the bounds (7) and (8) (note that q is at least 7). In Table 2 the parameters x and y are also shown. If we choose x = 0 and R(y) to be the bound (9) and we minimize the right-hand side of the inequality (12) (for minimization see [1], p.156), we obtain the bound (7). Our bound in Theorem 2 is analogous to the straight-line bound of [15], although its derivation is purely combinatorial. Table 1. Table 2.
8
Table 1. Numerical values for q = 16.
0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
(7) & (8) 0.70201 0.68056 0.65908 0.63755 0.61598 0.59435 0.57269 0.55098 0.52923 0.50744 0.48561 0.46374 0.44184
(12) 0.70192 0.67990 0.65788 0.63586 0.61384 0.59183 0.56981 0.54779 0.52577 0.50375 0.48173 0.45972 0.43770
0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76
(7) & (8) 0.41955 0.39722 0.37490 0.35257 0.33025 0.30792 0.28560 0.26327 0.24095 0.21862 0.19630 0.17397 0.15165
(12) 0.41568 0.39366 0.37164 0.34963 0.32761 0.30559 0.28357 0.26155 0.23953 0.21752 0.19550 0.17348 0.15146
Table 2. The interval [a,b] and parameters x and y.
a b x y q a b 7 0.54 0.56 0.08 0.58 27 0.14 0.84 8 0.52 0.60 0.08 0.62 32 0.12 0.86 16 0.26 0.76 0.06 0.78 64 0.06 0.92 q
x
y
0.04 0.86 0.04 0.88 0.02 0.94
4. New upper bounds for covering radius We are now in a position to state the results for covering radius. Theorem 3 is valid in the whole interval [0; 1] whereas Theorem 4 improves it in a certain part of this interval. Theorem 3 Let (Cn )1 n=1 be a sequence of nonbinary linear codes Cn of length n, 0 and covering dual distance d radius R where R=n ! and d0 =n ! 0 when n ! 1. ?
If 0 0 q?q 2 2 , then
1 ? 0 q?q 2 logq (q ? 1) p ; logq (1+1?00 )2
(13)
? and, if q?q 2 2 0 q?q 1 ,then
Hq(kq (p0 ))
logq
0 ?0
(1+ )2 1
:
Proof: We choose f (x) = tr (x), a = d0 and b = n. Then
(14)
9
max jf (x)j =
x [d0 ;n]
2
Thus f (0) +
n X i=1
Tr
n1 d0 +
n d0
?
1 ? (qn?k ? 1) i2max jf (i)j d0 ;n
i (b)f (i)
> 1?
Tr
qn?k
n+d0 n d0
?
[
(15)
]
:
Therefore, by Lemma 1, we have R r if qn?k Tr
n + d0 n ? d0
(16)
:
Combining the results (16) and (6) with the dual forms of the bounds (7) and (9); i.e., k lim sup n ? n n!1
1 ? 0 q ?q 2 logq (q ? 1); 0 0 q ?q 2
2
;
k Hq (kq (0 )); 0 0 q ?q 1 lim sup n ? n !1
n
gives the desired result. Note that the bounds (13) and (14) coincide at ((q ? 2)=q)2 . If we replace the dual forms of the bounds (7) and (9) with the dual form of the bound (12) where we have chosen R(y) = Hq (kq (y)); i.e., k Hq (kq (y)) ? 1 + Hq (x=2) 0 ( ? x) + 1 ? Hq (x=2) lim sup n ? n y?x !1
n
(17)
where 0 x 0 y q?q 1 , x 6= y, the same argument as in the previous proof gives the following result. We denote the right-hand side of (17) by Lx;y (0 ). n Theorem 4 Let (Cn )1 n=1 be a sequence of nonbinary linear codes Cn Fq with 0 0 0 dual distance d and covering radius R where R=n ! and d =n ! when n ! 1. Assume also that x 0 y, x 6= y where x; y 2 [0; (q ? 1)=q]. Then Lx;y (p0 ) : (18) logq (1+1?00 )2
10
In Figure 1 the comparison of the asymptotical results is shown for q = 16. We have used the label (16) for the bound obtained from (15) and when it is not valid (i.e. 0 > 0:78) from the bounds of Theorem 3. Notice that the bound (16) is general whereas the Delorme-Sole-Stokes bound uses additional conditions. It should be emphasized that in the binary situation further improvements are possible (see [8]).
Acknowledgements We are very grateful to M.Aaltonen for inspiring discussions, and for providing programs for calculating asymptotical bounds on the size of non-binary codes. The authors thank heartily also I. Honkala and A. Tietavainen for their valuable suggestions. We thank the referees for providing the references [3], [11] and [15]. Figure 1.
References 1. M. Aaltonen, Bounds on the information rate of a tree code as a function on the code's feedback decoding minimum distance, Ann. Univ. Turku. Ser. A I, Vol. 181 (1981). 2. M. Aaltonen, A new upper bound on nonbinary block codes, Discrete Math., Vol. 83 (1990), pp. 139{160. 3. F. R. K. Chung, V. Faber and T. A. Manteu el, An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian, SIAM J. Discr. Math., Vol. 7, (1994), pp. 443{457. 4. G. Cohen, S. Litsyn, A. Lobstein and H. Mattson, Jr., Covering radius 1985{1994. Technical Raport 94-D-025, Ecole Nationale Superieure des Telecommunications, Paris, 1994, submitted to AAECC Journal. 5. C. Delorme and P. Sole, Diameter, covering index, covering radius and eigenvalues,Europ. J. Combin., Vol 12 (1991), pp. 95{108. 6. P. Delsarte, Four fundamental parametres of a code and their combinatorial signi cance,Information and Control, Vol. 23, (1973), pp. 407{438. 7. T. Helleseth, T. Kl ve, and J. Mykkeltveit, On the covering radius of binary codes, IEEE Trans. Inform. Theory, Vol. 24 (1978), pp. 627{628. 8. I. Honkala, S. Litsyn, and A. Tietavainen, On algebraic methods in covering radius problems, In: Cohen, G., Giusti M., Mora, T. (eds.) Applied Algebra, Albegraic Algoritms and ErrorCorrecting Codes. Lecture Notes in Computer Science, Vol 948. Berlin, Heidelberg, New York: Springer (1995). 9. J. van Lint, Introduction to Coding Theory, Berlin, Heidelberg, New York: Springer-Verlag (1982). 10. S. Litsyn and A. Tietavainen, Upper bounds on the covering radius of a code with a given dual distance, Europ. J. Combin., to appear. 11. A. Lubotszky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica, Vol. 8, (1988), pp. 261{277.
11
12. F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes, Amsterdam: North-Holland (1977). 13. R. McEliece, E. Rodemich, H. Jr. Rumsey and L. Welch, New upper bounds on the rate of a code via Delsarte-MacWilliams inequalities, IEEE Trans. Inform. Theory, Vol. 23, (1977), pp. 157{167. 14. T. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons (1974). 15. C. E. Shannon, R. G. Gallager and E. R. Berlekamp, Lower bounds to error probability for coding on discrete memoryless channels. II, Information and Control, Vol. 10, (1967), pp. 522{552. 16. P. Sole, Asymptotic bounds on the covering radius of binary codes, IEEE Trans. Inform. Theory, Vol. 36, (1990), pp. 1470{1472. 17. P. Sole and K. Mehrotra, Generalization of the Norse bounds to codes of higher strenght, IEEE Trans. Inform. Theory, Vol. 37, (1991), pp. 190{192. 18. P. Sole and P. Stokes, Covering radius, codimension, and dual-distance width,IEEE Trans. Inform. Theory, Vol. 39, (1993), pp. 1195{1203. 19. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., v,. 23, Providence, RI, (1975). 20. A. Tietavainen, An upper bound on the covering radius as a function of its dual distance, IEEE Trans. Inform. Theory, Vol. 36, (1990), pp. 1472{1474. 21. A. Tietavainen, Covering radius and dual distance, Designs, Codes and Cryptography, Vol. 1, (1991), pp. 31{46.
