Linear programming bounds for. codes of small size. Ilia Krasikov. Tel-Aviv University. School of Mathematical Sciences,

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1 Linear programming bounds for codes of small size Ilia Krasikov Tel-Aviv University School of Mathematical Sciences, Ramat-Aviv Tel-Aviv, Israel and Beit-Berl College, Kfar-Sava, Israel Simon Litsyn Tel-Aviv University Department of Electrical Engineering { Systems Ramat-Aviv Tel-Aviv, Israel

2 Running head: Bounds for codes Mailing address: S.Litsyn, Department of Electrical Engineering - Systems, Tel-Aviv University, Ramat-Aviv Israel; tel.(973) (h), (973) (oce), (973) (fax); litsyn@eng.tau.ac.il Home page: litsyn

3 Abstract Combining linear programming approach with the Plotkin-Johnson argument for constant weight codes, we derive upper bounds on the size of codes of length n and minimum distance d = (n 0 j)=, 0 < j < n 1=3. For j = o(n 1=3 ) these bounds practically coincide (are slightly better) with the Tietavainen bound. For xed j and j proportional to n 1=3, j < n 1=3 0 (=9) ln n, it improves on the earlier known results. Keywords: Upper bounds, Plotkin bound, Tietavainen bound, McEliece bound. 3

4 1 Introduction Let C be a binary (n; M; d) code of length n, size M, and minimum distance d = (n0j)=. Let B = (B 0 ; B 1 ;... ; B n ) stand for its distance distribution. For every c C dene the relative weight distribution A(c) = (A 0 (c);... ; A n (c)) w.r.t. c, where A i (c) denotes the number of codewords of C being at distance i from c. So, B i = 1 M X cc A i (c): The MacWilliams transform of B is B 0 k = 1 M i=0 B i P k (i); and in every code B 0 = B 0 0 = 1, B 0 k 0 [3]. The tight bounds (provided existence of Hadamard matrices of all sizes divisible by 4) due to Plotkin and Levenshtein describe the best parameters of codes with j 0, and also j = 1, n = 3 mod 4 (see, e.g. [8, x.,.3]). Even in the case j = 1, n = 1 mod 4, the exact value of the maximum possible size of code is still unknown. In what follows, we deal with the case j > 0. In 1973 R.J.McEliece (unpublished, see [8, Theorem 38,Chapter 17]), using the linear programming approach has established the bound M < M ME := n(j + ); (1) valid for j = o( p n ). I.Honkala and A.Tietavainen [4] have constructed codes with parameters n = ( 4m 0 1)=( m + 1), j = m 0 1 and M = 4m, thus demonstrating that for j n 1=3 the McEliece bound is of the correct order of magnitude. We are not aware of any other general construction of codes with n 1=3 < j = o( p n), of size close to the McEliece bound. A. Tietavainen [16] developed an ingenious approach, resulting in an essential improvement on (1) for j = o(n 1=3 ). Namely, for xed j and for j growing with n, j = o(n 1=3 ), M < M T := (5 0 p + ln(j + 1))n; () n ln j M < MT 0 := (1 + o(1)): (3) Actually, the method of A.Tietavainen still yields meaningful results while j < n 1=3. 4

5 It is known [6, 1] that the direct use of the linear programming cannot lead to a better bound than that of McEliece. So, to progress one should use some extra ineualities. Here we will show that new ineualities naturally arise in the context of the Plotkin- Johnson arguments being used for estimating the size of codes in a Hamming ball. It turns out that this yields, for a wide class of polynomials (x), an upper bound on P ni=a (i)b i, where a > (n + p jn )=. On the other hand, the linear programming gives ineuality P n i=0 (i)b i M 0, where 0 = P 0n n n k=0 (k), whenever (x) has positive k coecients in its Krawtchouk expansion. Combining these two ineualities allows getting upper bounds on M. Here we apply this idea to specially chosen polynomials of second degree, and obtain bounds on the size of codes with j < n 1=3. The main result is as follows. Theorem 1 Size M of codes of length n and of minimum distance d = (n 0 j)=, 0 < j < n 1=3, is upperbounded as follows: i) j is xed, where " satises M < n (j ln( j 0 1) + 5j + 6") " + o(n); 4(1 0 ")j " (j ln(j 0 ") 0 j ln " + 5j + 6) ii) j is growing with n, n > j 3 ( ln j + 1), +"j(j ln " 0 j ln(j 0 ") 0 7j 0 6) + j = 0: M < n ln j( ln j + ln ln j + 5) 4(ln j 0 1) n ln j : + o(n) iii) j is growing with n, j 3 < n < j 3 ( ln j + 1), M < c n c (c 0 1) ln j (ln j + ln j ln ln j + ln ln j 0 c ln(c 0 1) + o(ln j ln ln j)) c 0 1 where c = p n j 03=. cn ln j c01 (c 0 1) ; 5

6 For xed j, the theorem gives M < const n with a better constant (depending only on j) then that in the Tietavainen bound M T (see Table 1 for the explicit values for rst few j's). Still, those constants can be further improved at the expense of more accurate calculations. For j = o(n 1=3 ), growing in n, our bound, although slightly better, practically coincides with MT 0. In view of the construction of Honkala and Tietavainen there is no hope to amend M ME for j n 1=3. Nevertheless, for j < n 1=3 0 ln n, our result still surpass the 9 McEliece bound. In the proof we use linear programming with polynomials of second degree. Analogous machinery can be applied to polynomials of higher degrees. Apparently, it can lead to an improvement on the Levenshtein bounds [6], see also []. The Tietavainen bound was generalized to the case of nonbinary [10], constant weight [15], Lee [15], and spherical [1] codes. Clearly our results can be expanded also to these situations. Everywhere in the paper we assume n to be suciently large to justify all approximations. Ineualities We start with the auxiliary lemma which employs the well known Johnson-Plotkin argument usually being used for constant weight codes. However, and it is crucial for our purposes, the same arguments are valid as well for estimating the maximal size of a code of given distance within (or outside) a Hamming sphere. We present here the sought modication of the standard proof for this situation (see also [9]). Lemma 1 For every c C, ( i=x and, if x > (n + p jn )=, (i 0 n)a i (c)) nm c (x)((m c (x) 0 1)j + n); (4) m c (x) = i=x A i (c) dn x 0 nx + dn : (5) Proof. Fix c C. Consider the matrix V 6 consisting of the codewords being at distance at least x from c with zeros replaced by (01)'s. The number of its rows is precisely m = m c (x) = P n i=x A i (c). 6

7 Let v 1 ;... ; v m ; stand for the rows of V 6, while u 1 ;... ; u n, being its columns. Denote also by s i the number of 1's in the i-th column, and by s the average number of 1's in column, s = 1 s i = 1 n n i=1 k=x ka k (c): (6) The usual scalar product of any two distinct rows is at least n 0 d = j, therefore S = mx mx i=1 k=1 On the other hand, by Jensen ineuality, (v i ; v k ) m(m 0 1)j + mn: So, S = i=1 s i + (m 0 s i ) 0 s i (m 0 s i ) = i=1 (s i 0 m) n(s 0 m) : n(s 0 m) m(m 0 1)j + mn; that is euivalent to (4). Finally, since s xm=n, solving the ineuality we get (5) provided x > (n + p jn )=. We need the following elementary technical lemma. Lemma The maximum in z, z > 0, of the function u(z) = nz((z 0 1)j + n) 0 wz; euals (n 0 j)(w 0 p w 0 jn) ; j provided w p jn, and u(z) is increasing in z if jwj < p jn. Proof. We have u 0 (z) = n(n 0 j + jz) nz(n 0 j + jz) 0 w; (n 0 j) nz(n 0 j + jz) u 00 (z) = 0 < 0: 4z (n 0 j + jz) 7

8 For jwj < p jn we have u 0 (z) > 0. For w p jn, the maximum is achieved at z = (n 0 j)(w 0 p w 0 jn) j p : w 0 jn Here follows the theorem presenting the key ineuality. Theorem Let x (n + r)=, r > p jn, then 8 >< >: where y = x 0 n 0 r+j. i=x n(n0j)(r+j) 4((x0n) 0jn) n0j 4j (y 0 p y 0 jn) (i 0 x + r + j 4 )B i g(x) := n 0 x + r+j 4 if 4x > 3n + j if (n + r) < 4x < n + r + j + p jn if n + r + j + p jn 4x 3n + j (7) Proof. Let c be a codeword. Observe that for x > n= we have s 0 m = i=x Therefore, using d = (n 0 j)=, from (4) we derive This is euivalent to Adding r+j i=x n(s 0 m) = i=x i=x (i 0 x)a i (c) i n 0 1 A i (c) 0: (i 0 n)a i (c) P ni=x A i (c) = r+j m, we obtain i 0 x + r + j 4 nm((m 0 1)j + n) : nm((m 0 1)j + n) 0 (x 0 n)m: A i (c) nm((m 0 1)j + n) 0 x 0 n 0 r + j m: If 4x n + r + j + p jn, the previous lemma, with w = x 0 n 0 r+j z = m, gives the maximum of the right-hand side in m. p jn and 8

9 If (n+r) < 4x < n+r+j+ p jn, then jwj < p jn, and the maximum of the right-hand side is attained for the maximal possible value of m given by (5). Finally, for x > 3n+j 4, there is at most one codeword of weight being in the range [x; n]. Therefore, for such x, i=x A i (c) i 0 x + r + j 4 max i i 0 x + r + j 4 = n 0 x + r + j 4 : Averaging over all c C we get the claim. Lemma 3 Let (x) be a nonnegative polynomial for all integer x in the interval [(n + r)=; n], r > p jn. Then i=(n+r)= B i ix x=(n+r)= (i 0 x + a) (x) x=(n+r)= g(x) (x): (8) Proof. Multiplying both sides of (7) by (x) and summing over x [(n + r)=; n], we obtain the claim changing the order of summation in the left-hand side. Remark 1 Although, in what follows, we use (x) = 1, for j being greater than n 1=3 one should use polynomial (x) of larger degree. Theorem 3 For r > p jn, y = x 0 n 0 r+j, i=(n+r)= (n + r 0 i 0 )(n 0 j 0 i) B i 8 x=(n+r)= g(x) (9) = 8 >< >: 1 P (3n+j)=4 (n 0 j) j (y 0 p y 0 jn) + (n + j + r) x=(n+r)= + 1 j (n 0 j) n(r + j) P (n+r+j+ p jn )=4 x=(n+r)= 1 (x0n) 0jn P (3n+j)=4 x=(n+r+j+ p jn )=4 (y 0 p y 0 jn) + (n + j + r) for r j + p jn for r < j + p jn Proof. Follows from Theorem and Lemma 3 with (x) = 1. 9

10 In the following lemma we give an ineuality slightly more general than that of the conventional linear programming [3] if one uses polynomials of second degree. Generalization to arbitrary degree is straightforward. Lemma 4 Let f(x) = (x 0 x 1 )(x 0 x ) = a 0 + a 1 P 1 (x) + 1 P (x); where a 0 ; a 1 0, x x 1 d. Then, provided a 0 > f(d), M f(0) + P n i=x f(i)b i : (10) a 0 0 f(d) Proof. From the expression of the MacWilliams transform Ma 0 M(a 0 B a 1B (1=)B0 ) = f(0)b 0 + Xx 1 i=d i=0 f(i)b i + f(i)b i i=x f(0) + Mf(d) + f(i)b i : i=x f(i)b i It is easy to see that if x 1 d and x n, then we get the standard linear programming bound: M f(0)=a 0. Theorem 4 M < inf r( p jn ;+n=j) (n 0 j)(n + r 0 ) + 8 P n x=(n+r)= g(x) : n 0 jr + j Proof. Choose in the previous lemma f(x) = 1 4 (n 0 j 0 x)(n + r 0 x 0 ) = (n 0 jr + j) + (r 0 j 0 )P 1(x) + P (x) : 4 Moreover, x 1 = d, and so f(d) = 0. To have a 0 and a 1 nonnegative along with r > p jn we reuire 10

11 jn < r < n j + : (11) Substituting the chosen f(x) into (10) we get M (n 0 j)(n + r 0 ) + P n x=(n+r)= (n 0 j 0 x)(n + r 0 x 0 )B x n 0 jr + j (n 0 j)(n + r 0 ) + 8 P n x=(n+r)= g(x) : (1) n 0 jr + j 3 Specic bounds To derive an explicit upper bound we optimize in r the ineuality of Theorem 4. It turns out that the main term of the bound is not essentially sensitive to the choice of r. Consider several cases depending on the order of j. Case j is a constant independent on n In this case we choose r = "n=j, for some constant " = "(j) > 0, to be chosen later. Then p jn = o(r), and Using 1 0 p 1 0 z z= 0 z =; g(x) 4x n + r > n + r + j + jn : 8 >< >: z [0; 1]; we get n(n0j) 8y where y = x 0 n 0 r+j. This gives + n(n0j)j 8y 3 for 4x 3n + j n 0 x + r+j 4 for 4x > 3n + j M < (n 0 j)(n + r) + n(n 0 j) P (3n+j)=4 x=(n+r)= (y01 + jy 03 ) + 8 P n x=(3n+j)=4 (n 0 x + r+j ) 4 n 0 jr + j < n 5n + 6r + n ln n0r r 4(n 0 jr) + o(n): 11

12 Table 1: Bounds for j xed j " M=n M T =n M ME =n 1 0: :7585 : : : : : :8887 3: : :4459 3: : : : : : : :185 3: : : :881 4: : :8880 4: : : : : :704 6: To nd the minimum of the right-hand side we determine (uniue) " [0; 1], from the condition: " (j ln(j 0 ") 0 j ln " + 5j + 6) +"j(j ln " 0 j ln(j 0 ") 0 7j 0 6) + j = 0: For rst few j's the resulting upper bound is presented in Table 1 (here M stands for the suggested bound, M T for the Tietavainen bound, and M ME for the McEliece bound). Remark Actually, the Plotkin bound for j = 1 and n = 3 mod 4 yields M n +, and this bound is achieved. For j = 1 and n = 1 mod 4, the upper bound M :7585:::n + o(n), seems to be the best known. The conference-matrix codes in this case give M = n + [13]. Remark 3 For small j, the bound we propose can be further tightened. It can be done if in Lemma 1 we take into account that the number of 1's and (-1)'s in each column is integer (see [5]). For instance, there can be at most four codewords of weight greater than 0:7n + o(n), ve of weight greater than (5=36)n + o(n), etc. Exact calculations reuire a vast amount of computational work, and apparently lead to marginal improvements. Case j is growing, n > j 3 ( ln j + 1) Choose r = n. Then, again, j ln j 1

13 4x n + r > n + r + j + jn ; and proceeding as in the previous case, we get M < n ln j( ln j + ln ln j + 5) 4(ln j 0 1) + o(n): n ln j (1 + o(1)): Case j is growing, j 3 < n < j 3 ( ln j + 1) Let n = c j 3, where 1 < c < ln j + 1. Set r = p jn, where 1 < = 1 + c 0 1 ln j < n + j j p jn : (13) Then M < 1 n 0 jr 1 (n 0 j)(n + r) + n(n 0 j)(r + j) (n+r+j+ X p jn )=4 = 1 n 0 jr (n 0 j) + j +8 (3n+j)=4 X x=(n+r+j+ p jn )=4 (3n+j)=4 (n 0 j) j +8 (3n+j)=4 x=(n+r)= (y 0 1 (n 0 x + r + j 4 ) A (3n+j)=4 X x=(n+r+j+ p jn )=4 (n 0 x + r + j 4 ) y 0 jn) (y 0 1 A 1 (x 0 n) 0 jn y 0 jn) = n 0 j n 0 jr (5n + 6r + I 1 + I ); (14) 13

14 where (n+r+j+ p jn )=4 X 1 I 1 = n(r + j) (x 0 n) 0 jn ; x=(n+r)= I = j (3n+j)=4 X x=(n+r+j+ p jn )=4 (y 0 y 0 jn): We estimate now I 1 and I replacing the sums by integrals. We get I 1 = s n j (j + r) arth r p 0 arth j + r +! p jn jn p + o(n) jn = p n ( p n + j ) 0 arth 0 arth 1 j 1 + n + 11 A A + o(n) = 1 p p ( + 1)( p j + p n ) n ( n + j ) ln ( 0 1)( p j + ( + 4) p n ) + o(n) Similarly, = n ln ( + 1) ( 0 1)( + 4) + o(n): I = 1 8j 1 0 (n 0 r) 0 4jn 0 (n 0 r) (n 0 r) 0 4jn + 4jn ln n 0 r + (n 0 r) 0 4jn p A + o(n) jn = n(4jn 0 (n 0 r) ) 4((n 0 r) 0 3jn) + n (n 0 r) ln + o(n) 4 jn = n 4 ( ln(p n 0 j ) 0 ln j 0 1) + o(n) = n 4 ln n j 0 1! + o(n): 14

15 Now (14) gives M < n 0 j n 0 j p jn 5n + 6 j jn + n ln ( + 1) ( 0 1)( + 4) + n 4 (ln n j 0 1) + o(n)! = n(n 0 j) 4(n 0 jr) ln n j 0 ln ( 0 1)( + 4) ( + 1)! o(1) Substituting from (13) and simplifying we get: M < c(n 0 j) c ln j + ln j ln ln j + ln (c 0 1) ln j c 0 1 ln j 0 c ln(c 0 1) + o(ln j ln ln j) cn ln j c01 (c 0 1) : Notice that this bound is better then the McEliece bound while j < n 1=3 0 9 ln n+o(ln n): 4 Acknowledgement The rst author is thankful to Gerard Cohen for providing hospitality while working on the paper. References [1] J.T.Astola, The Tietavainen bound for spherical codes, Discr. Appl. Math. 7 (1984), 17{1. [] P.Boyvalenkov, preprint (1996). [3] P.Delsarte, An algebraic approach to association schemes of coding theory, Philips Research Rep. Supp., 10 (1973). [4] I.Honkala and A.Tietavainen, Codes and number theory, in: Handbook of Coding Theory, Elsevier, to be published. [5] S.M.Johnson, A new upper bound for error-correcting codes, IRE Trans. Info. Theory, 8 (196), 03{07. 15

16 [6] V. Levenshtein, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Trans. Info. Theory, 41 (1995), pp.1303{131. [7] J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, 199. [8] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, New York: North-Holland, [9] R.J. McEliece and H.Rumsey,Jr., Sphere-packing in the Hamming metric, Journal of Combinatorial Theory, Ser.A (1973), 3{34. [10] A.Perttula, Bounds for binary and nonbinary codes slightly outside of the Plotkin range, Tampere Univeristy of Technology Publ., 14 (198). [11] M.Plotkin, Binary codes with specied minimum distance, IRE Trans. Info. Theory, 6 (1960), 445{450. [1] V.M.Sidel'nikov, On extremal polynomials used to estimate the size of codes, Probl. Inform. Transmission, 16 (1980), 174{186. [13] N.J.A.Sloane and J.J.Seidel, A new family of nonlinear codes obtained from conference matrices, Annals N.Y. Acad. Sci., 175 (1970), 363{365. [14] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Collo. Publ., vol.3, Providence, RI, [15] H.Tarnanen, Upper bounds for constant weight and Lee codes slightly outside the Plotkin range, Discrete Applied Mathematics, 16 (1987), 65{77. [16] A.Tietavainen, Bounds for binary codes just outside the Plotkin range, Information and Control, 47 (1980), 85{93. 16

Improved Upper Bounds on Sizes of Codes

880 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 4, APRIL 2002 Improved Upper Bounds on Sizes of Codes Beniamin Mounits, Tuvi Etzion, Senior Member, IEEE, and Simon Litsyn, Senior Member, IEEE