Bounds for binary codes with narrow distance distributions

Transcription

1 Bounds for binary codes with narrow distance distributions Ron M Roth, Gadiel Seroussi Advanced Studies HP Laboratories Palo Alto HPL September 9, 006* constant-weight codes, distance distribution, equidistant codes, linear programming bound New lower bounds are presented on the second moment of the distance distribution of binary codes, in terms of the first moment of the distribution These bounds are used to obtain upper bounds on the size of codes whose maximum distance is close to their minimum distance It is then demonstrated how such bounds can be applied to bound from below the smallest attainable ratio between the maximum distance and the minimum distance of codes Finally, counterparts of the bounds are derived for the special case of constant-weight codes * Internal Accession Date Only Approved for External Publication Copyright 006 Hewlett-Packard Development Company, LP

2 Bounds for binary codes with narrow distance distributions Ron M Roth Gadiel Seroussi September 18, 006 Abstract New lower bounds are presented on the second moment of the distance distribution of binary codes, in terms of the first moment of the distribution These bounds are used to obtain upper bounds on the size of codes whose maximum distance is close to their minimum distance It is then demonstrated how such bounds can be applied to bound from below the smallest attainable ratio between the maximum distance and the minimum distance of codes Finally, counterparts of the bounds are derived for the special case of constant-weight codes Keywords: Constant-weight codes; Distance distribution; Equidistant codes; Grey Rankin bound; Linear programming bound; Quasi-symmetric designs; Selfcomplementary codes 1 Introduction Let C be an n, M, d) binary code of length n, size M > 1, and minimum Hamming distance d) The distance distribution of C is a list B 0 B 1 B n ), where B i is the average over all This work was done in part while both authors were at Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, CA 930, USA This work was supported in part by grant No from the United- States Israel Binational Science Foundation BSF), Jerusalem, Israel This work was presented in part under a different title) at the IEEE International Symposium on Information Theory ISIT 005), Adelaide, Australia September 005), pp in the proceedings Computer Science Department, Technion, Haifa 3000, Israel ronny@cstechnionacil Mathematical Sciences Research Institute MSRI), 17 Gauss Way, Berkeley CA 970, USA gadiel@msriorg 1

3 codewords c C of the number of codewords at Hamming distance i from c; that is, if d, ) denotes Hamming distance, then B i = 1 {c C : dc, c ) = i}, 0 i n M Equivalently, c C Clearly, B 0 = 1 and B i = 0 when 0 < i < d B i = 1 M {c, c ) C C : dc, c ) = i} Define β i = B i /M for 0 i n Then β i ) n is the probability measure on the distances in C that is induced by assuming a uniform distribution on the codewords of C In particular, E C = iβ i is the average distance between any two codewords in C, and S C = i β i is the second moment of the distances It is known that E C is bounded from above by n/ in fact, this is the basis of the proof of the Plotkin bound: see [9, pp 1 ]) In this work, we obtain a new lower bound on the second moment S C, in terms of the code parameters n and M and the average E C This bound will be presented in Section 3 and a counterpart of that bound for constant-weight codes will be presented in Section 5) There are known bounds on the moments of codes, obtained from MacWilliams identities, the Delsarte linear programming DLP) bound, and enhancements thereof: see [1], [], [3], [9, Sections 5 and 17] We show that there is a range of code parameters for which the bound we present here is not implied by these techniques As an application of our new lower bound, we present in Section an upper bound on the size M of a binary code C, in terms of its length n, minimum distance d, and the maximum distance d max between any two codewords in C The effectiveness of our bound can be seen in cases where d max is close to d, ie, when the distance distribution of the code is required to be narrow As a concrete practical motivation for studying such codes, we refer the reader to [8], where it is demonstrated how binary constant-weight codes with narrow distance distribution can be incorporated into the design of demultiplexers for nano-scale memories The following definition will be useful throughout this work Let A = AC) be an M n real matrix whose rows are indexed by the codewords of C, and the row of A that is indexed by c is given by A) c = c 1 ;

4 here 1 denotes the real all-one vector of length n and the codeword c is regarded as an integer vector over {0, 1} n thus, the entries of A are in { 1, +1}) For i = 0, 1,, n, denote by E i the M M matrix whose rows and columns are indexed by the codewords of C, and for every c, c C, { 1 if dc, c E i ) c,c = ) = i 0 otherwise Let J denote the M M all-one matrix Clearly, we have E 0 = I and n E i = J It follows that AA T = n i) E i = n J i E i 1) Example 11 An n, M, d) binary code C is equidistant if B i 0 only if i {0, d} For equidistant codes we have I if i = 0 E i = J I if i = d 0 otherwise and, so, i=1 AA T = n d) J + d) I ) Bound on the size of equidistant codes In this section, we present an upper bound on the size of equidistant codes At first sight, this result might not seem to be related to the problem of bounding the second moment of the distances However, the proof technique that we use here bears a common ground with the analyses in subsequent sections, through the properties of the matrix AA T in 1) Recall that the Plotkin bound states that for every n, M, d) binary code C for which n < d, M d d n, and equality can be attained only if C is equidistant see [9, pp 1 ]) Proposition 1 Let C be an equidistant n, M, d) binary code Then M n+1, and equality holds if and only if n = d 1 and C attains the Plotkin bound Proof The if part is straightforward: when n = d 1 and C attains the Plotkin bound, then M = d = d = n+1 d n 3

5 Turning to the only if part, the proof is similar to that of Fisher s inequality in block designs see [, p 81] or [9, p 6]) Specifically, we compute the rank of the matrix AA T in two ways On the one hand, rankaa T ) = ranka) n 3) On the other hand, we see from ) that the eigenvalues of AA T multiplicity 1) and d with multiplicity M 1) Therefore, are d + Mn d) with { M 1 if d + Mn d) = 0 rankaa T ) = M otherwise ) We conclude from 3) and ) that M 1 n, with equality holding only if d+mn d) = 0 In case of equality we have n+1 = M = d d n, and it can be readily seen that d/d n) may exceed n only when n = d 1 The bound in Proposition 1 is attained by Hadamard codes [9, Section 3] The n>3, M=n, d=) code that consists of all binary words of length n and Hamming weight 1 is equidistant and its size is the largest possible for its length, given that n > d 1 See Tonchev [11, Section ] and the references therein) for conditions on the existence of equidistant codes that meet the Plotkin bound A characterization of all linear equidistant codes can be found in Bonisoli [5] 3 Bound on the second moment For a real M M matrix Y = y i,j ) M i=1 M j=1, denote by TrY ) its trace and by Y F Frobenius norm, ie, M M 1/ Y F = yi,j) It is easy to verify that i=1 j=1 Y F = TrY ) We make use of the following theorem, which shows the effect on the eigenvalues of a given real symmetric matrix X, caused by subtracting from X a nonnegative definite symmetric matrix of rank 1 see Wilkinson [1, pp 9 97]) Theorem 31 Let X be a real symmetric M M matrix with eigenvalues λ 1 λ λ M and let v be a real row vector of length M Define the matrix Y by Y = X v T v, its

6 and let µ 1 µ µ M be the eigenvalues of Y Then, for r = 1,,, M 1, λ r+1 µ r λ r, and µ M λ M Proposition 3 Let M β i ) n be the distance distribution of an n, M n, d) binary code Then for every nonnegative real γ n/, Proof From 1) we have, i γ) β i γ n AA T = n γ) J + γ i) E i and, so, M i γ) β i = γ i) E i F = AA T n γ) J F 5) In what follows, we bound from below the right-hand side of 5) To this end, we apply Theorem 31 to the symmetric matrix X = AA T and the vector Then, v = n γ ) Y = AA T n γ) J, 6) and the right-hand side of 5) equals Y F = TrY ) Note also that TrY ) = Mγ The matrix X = AA T is nonnegative definite ie, λ M 0), and rankx) = ranka) n; hence, λ n+1 = λ n+ = = λ M = 0 Therefore, by Theorem 31, the eigenvalues of Y satisfy µ n+1 = µ n+ = = µ M 1 = 0 7) and µ M 0 8) 5

7 From 7) 8) it follows that µ r r=1 M µ r = TrY ) = Mγ 0) r=1 Thus, Y F = TrY ) = M µ r r=1 µ r 1 ) M γ µ r n n r=1 where the penultimate inequality follows from the convexity of z z ) The result is now obtained by combining 5), 6), and 9) r=1 9) Theorem 33 Let C be an n, M n, d) binary code Then ne C if E C S C ne C n) if < E C n Proof We observe that i γ) β i = S C γe C + γ, and by Proposition 3 we thus get S C γe C n γ, 10) for every real γ n/ The right-hand side of 10) attains its maximum at ne C if E C γ = n if < E C n, and the result is obtained by substituting this maximizing value of γ into 10) We point out that the inequality S C ne C n) 11) 6

8 obtained by substituting γ = n/ in Proposition 3) is implied by the DLP bound To see this, recall that MacWilliams identities relate the distance distribution B i ) n to its MacWilliams transform B l )n l=0 by B l = 1 M K l i; n) B i, 0 l n, 1) where K l x; n) = l ) ) x n x 1) j, 0 l n j l j j=0 The DLP bound on the code size is then given as the largest attainable value of the sum B 0 + B B n = M), where B i ) n ranges over all nonnegative rational vectors whose MacWilliams transform satisfies the following constraints: B l 0, 0 l n 13) strictly speaking, the rational entries B i should also be such that their sum M is one of their integer common denominators) For l = we have ) n K x; n) = x nx + and, so, thereby yielding 11) 0 B = S C ne C + ) n, From this analysis it follows that when E C )/ eg, when C is a linear code with no all-zero coordinates), Theorem 33 offers no improvement over the DLP bound However, as we see in the sequel, there are instances where Theorem 33 implies a stronger bound than the DLP bound Note that Theorem 33 is at least as strong as Proposition 3, for any γ n/: Proposition 3 is equivalent to the linear constraint 10) on S C and E C for any fixed γ, while Theorem 33 is the intersection or the envelope) of the constraints 10) over all γ n/ The advantage in having the individual linear constraints 10) available is that they can be incorporated into any bounding technique that uses linear programming A qualitative) illustration of the bound of Theorem 33 is shown Figure 1 The curve changes from quadratic to linear in E C at the point E C, S C ) =, n) ): note that it is the quadratic region where the theorem strictly improves on the DLP-implied bound 11) 7

9 S C n, nn+1) ), n) ) Theorem 33 Equation 11) E C E C 1 Figure 1: Bounds on S C in terms of E C E C The figure is, in fact, drawn to scale for the length value n = 5; as such, it exaggerates the proportions of the linear region of the bound For larger n, the proportions change in favor of the quadratic region) For reference, we have also included in the figure the curve E C E C : the inequality S C E C must hold just from the fact that β i) n is a probability measure Bounds for narrow distance distributions Theorem 33 can be used to obtain bounds on the parameters of codes whose distance distributions are limited to a narrow range of distances Specifically, let C be an n, M, d) binary code and denote by d max the maximum distance between any two codewords in C We will be interested in cases where d max is close to d Equidistant codes are an extreme case where d max = d The next proposition yields an upper bound on the code size M, in terms of n, d, and d max 8

10 Proposition 1 Let C be an n, M, d) binary code with maximum distance d max Denote by d a and d g the arithmetic and geometric means, respectively, of d and d max ; ie, Then 1 1 M d a = d max + d 1 1 ) da ) n d g n d a n+1 ) d g and d g = d max d if d a {n+1, n+} if d a {n+1, n+} Proof For d max = d, the result follows from Proposition 1 In addition, the result can be easily verified to hold for the values n =, d = 1, and d max = Therefore, we assume hereafter in the proof that n 3 and d max > d For fixed value of E C, the second moment S C attains its maximum when the distance distribution is concentrated at the boundary distance values, ie, when β i 0 only for i {0, d, d max } In our proof, we apply Theorem 33 to this extreme worst case) distribution Assuming such a distribution, write v = β dmax ; then β d = 1 1/M) v and, so, E C = d 1 1 ) M v + d max v = d 1 1 ) + d max d)v, M or Similarly, d max d)v = E C d 1 1 ) 1) M S C = d 1 1 ) M v + d max v = d 1 1 ) + d max M d )v = d 1 1 ) + d max + d) M = d a E C d g 1 1 ), M E C d 1 1 ) ) M where the third equality follows from 1) We now distinguish between two cases Case 1: E C )/ By Theorem 33 we have ne C d ae C + d g 1 1 ) = ne C M S C 0 15) 9

11 This inequality has a solution for E C only if the discriminant of the quadratic expression in E C ) in 15) is nonnegative, namely, which is the same as or d a d g n 1 1 ) 0, M 1 1 M 1 1 ) da ) 16) n d g Case : E C > )/ We again apply Theorem 33 to obtain ne C n) d a E C + d g 1 1 M 1 d g 1 1 ) = ne C n) S C 0, M d a n)e C + n) ) 17) For d a n/, we bound from above the right-hand side of 17) by replacing E C with )/; this yields 1 1 M 1 d d a ) n) ) 18) g Similarly, for d a > n/, we get the next bound from 17) by substituting E C for n/: 1 1 M 1 d d a n nn+1) ) 19) g A simple check reveals that the right-hand side of 18) is never greater than the righthand side of 16); this means that the bound 16) prevails for d a n/ On the other hand, the right-hand side of 19) exceeds that of 16) whenever n < d a < nn+1) ; 0) observing that n+ < nn+1)/) n+3 for n 3, we conclude from 0) that when d a {n+1, n+}, the upper bound on 1 1/M) is dictated by 19) Remark Since the last proof is based on putting the distribution mass on the extreme distance values, Proposition 1 is useful when d max is bounded away from n; otherwise, our strategy would be too pessimistic For example, it is easy to see that in every n, M, d) binary code, the tail of the distance distribution satisfies B i 1, i>n d/) 10

12 and taking this inequality into account, our bound can then be improved when d max > n d/) We next present two applications of Proposition 1; both take advantage of the fact that the proposition yields the strongest bound when the distance distribution is two-valued ie, the distribution is nonzero only at two nonzero distance values) The first application is an alternate proof of the Grey Rankin bound see [7] and [9, p 5]) Theorem The Grey Rankin bound) Let C be an n, M, d) binary code that is selfcomplementary, ie, if c is a codeword in C then so is the word that is obtained by changing each 1 in c into 0 and each 0 into 1 If n n)/ < d n/ then M 8dn d) n n d) Proof Let C be the n, M/, d ) code obtained by taking one codeword of C from each complementary pair The minimum distance d of C is at least d and its maximum distance d max is at most n d Let d a and d g be given by 1 d max + d ) and d g = d max d, respectively, and define the function z Qz) over the positive reals by Qz) = 1 z + 1 ) z + It is easy to verify that da ) d ) d ) = Q Q = d g d max n d n dn d), where the inequality follows from the fact that Qz) is a decreasing function for z 0, 1] By Proposition 1 we thus have 1 M 1 1 ) da ) 1 1 ) d ) Q = n) n d g n n d dn d), or The result follows ) M n n d) 8dn d) In our next application of Proposition 1, we consider the case where the equidistant property is slightly relaxed As is commonly done in bounds that are based on the distance distribution, we will limit ourselves to even-distant codes, namely, B i 0 only if i is even when d is even, this condition can be guaranteed simply by changing the last coordinate in each codeword of C into a parity bit; when d is odd, we apply the bounds to the n+1, M, d+1) code obtained by adding a parity bit as an n+1)st coordinate) An n, M, d) binary code C is called nearly-equidistant if it is even-distant and d max = d + 11

13 Proposition 3 Let C be a nearly-equidistant n, M, d) binary code Then the following holds i) If n d n/) 1 then M dd+)n d+1) n ii) If n/) 1 < d n+1)/ then n+3))/n 3) if d = )/ M n+ if d = n/ n+1 if d = n+1)/ Proof Part i) corresponds to the case where d a < n+1 in Proposition 1 Part ii) corresponds to the case where either d a {n+1, n+} or the Plotkin bound is attained Example 1 For n = 13 and d =, Proposition 3i) yields the upper bound M 6 This bound is attained by a certain 13, 6, ) binary code whose codewords all have Hamming weight 3 see Brouwer et al [6, Table I-A]); thus, this code is nearly-equidistant In comparison, the DLP bound yields M 0 for these parameters with a feasible solution B = 13 and B 6 = 6) Example The linear n=9, M=, d=) binary code that is generated by the matrix G = can be verified to be nearly-equidistant with minimum distance d = )/ = This code attains the bound in Proposition 3ii), which coincides with the DLP bound in this case Example 3 As shown in [9, p 59], by puncturing the binary simplex code, one can obtain linear nearly-equidistant n= m, M= m, d= m 1 ) binary codes; these codes attain the bound in Proposition 3ii), which again, as expected, coincides with the DLP bound For the code in Example 1 we have d a {n+1, n+}, while the codes in Examples and 3 satisfy d a {n+1, n+} The former therefore belongs to the first range of values of d a in Proposition 1, while the latter belong to the second range therein Each of these 1

14 examples attains the bound in Proposition 1 and violates the bound that corresponds to the range that it does not belong to This shows that the distinction between the two ranges of values of d a in Proposition 1 is necessary Table 1 shows the lower bounds on the length n, which are computed from Proposition 3 and the DLP bound, respectively, for nearly-equidistant n, M, d) binary codes with M = 6 and d =,, 6, 8, 10 the value 6 for M was selected to match the example in [8, Section ]) Note that for d =, the DLP bound yields a better bound than Proposition 3 d Proposition 3 DLP Bound Table 1: Lower bounds on the length n of nearly-equidistant n, M=6, d) binary codes There are coding applications where the parameters of interest are the code length n, size M, and the ratio d max /d rather than the specific values of d and d max ) One such application is the design of demultiplexers for nano-scale memory platforms, where M stands for the memory size and n is the number of address lines, and the goal is to have d max /d as small as possible [8] in this application, the codes are also required to be of the constant-weight type; we consider such codes in Section 5) Propositions 1 and 3 can be used to find a trade-off between the parameters n, M, and d max /d for even-distant codes, as demonstrated in the next example Example We find the largest even-distant binary code with n = 7 and d max /d = /3 From Proposition 3 it follows that every nearly-equidistant n=7, M, d=6) binary code has size M 58 in comparison, the DLP bound provides a feasible solution for M = 309) We next check the values d = 1 and d max = 16 In this case, either bound Proposition 1 or the DLP bound implies the upper bound M 6 There actually exists a linear n=7, M= 6, d=1) code with d max = 16: it is obtained by shortening a selfcomplementary linear 8, 7, 1) code which attains the Grey Rankin bound see Parker et al [10] and Tonchev [11, p 163]) The ratio d max /d = /3 can also be realized by d = 18 and d max =, yet by the Plotkin bound, the largest 7, M, 18) binary code has size M = Using Propositions 1 and 3 and the DLP bound combined), we have created Table : this table provides a lower bound on d max /d as a function of n, where we have taken the size M to be 6 13

15 Range of n d max /d 7 n n 13 1 n 17 5/3 18 n 3/ 3 n 6 7/5 7 n 3 /3 Table : Lower bounds on the ratio d max /d for even-distant n, M=6) binary codes, as a function of n The lower bounds in Table are known to be tight, except possibly for the penultimate row For n = 7 and therefore for every n 7), the ratio d max /d = 3 is attained by the 7, 6, ) parity code for n > 7 we just pad each codeword with a tail of n 7 zeros), and a ratio of d max /d = is attained by the 11, 7, ) code defined in [9, pp 70 71] By shortening the 16, 56, 6) Nordstrom Robinson code, we get a code that attains a ratio of 5/3 for n = 1 see [9, pp 73 7]); similarly, by shortening the extended binary Golay code one can obtain a linear 18, 6, 8) code with a ratio of 3/ Finally, the code described in Example attains a ratio of /3 for n = 7 As of yet, we do not know whether n = 3 is the smallest length for which a ratio of 7/5 is achievable 5 The constant-weight case An n, M, d) binary code C is called a constant-weight code if all the codewords in C have the same Hamming weight We then say that C is an n, M, d, w) code, where w is the Hamming weight of each codeword There is no loss of generality in assuming that w n/, and we will indeed assume this inequality throughout this section Constant-weight codes are always even-distant Proposition 3 can be slightly improved in the case of constant-weight codes, as we show next Proposition 51 Let M β i ) n be the distance distribution of an n, M n, d, w) code, and denote by θ the ratio w/n Then for every nonnegative real γ θ1 θ)n, i γ) β i γ Proof The proof is similar to that of Proposition 3, except that we re-define the M n 1

16 matrix A so that its rows are now given by A) c = c θ 1) the multiplier in this definition is not essential to derive our results: it is inserted only to make the definition here closer to the one in Section 1) It follows that for every codeword c C, A) c 1 T = c 1 T ) }{{} w θ 1 1 T ) = 0 }{{} n This implies that A1 T = 0, which readily means that ranka) Also, for any two codewords c and c, A) c A) T c = c θ 1) c θ 1) T c 1 T ) = c c T ) θ }{{} w dc,c ) } {{ } w = θ1 θ)n dc, c ) + 1 c T ) }{{} w ) θ 1 1 T ) }{{} n Hence, the counterparts of) Equations 1) and 5) become, respectively, and Here M AA T = θ1 θ)n i) E i = θ1 θ)n J i γ) β i = Next, we apply Theorem 31 to X = AA T and F i E i γ i) E i = AA T θ1 θ)n γ) J 1) F v = θ1 θ)n γ ) Y = AA T θ1 θ)n γ) J and, since rankx) = ranka), we get that ie, λ n is also zero); thus, 7) becomes and 8) still holds It follows that λ n = λ n+1 = = λ M = 0 µ n = µ n+1 = = µ M 1 = 0, i=1 µ r r=1 M µ r = TrY ) = Mγ 0) r=1 15

17 and, so, Y F = TrY ) r=1 µ r 1 Finally, we combine the latter equation with 1) r=1 µ r ) M γ Theorem 5 Let C be an n, M n, d, w) code and define α = αn, w) by α = w 1 w ) n Then S C n E C α E C n )α if E C n )α if n )α < E C α Proof Similarly to 10), we get from Proposition 51 that γ = S C γe C n γ, ) for every real γ α The right-hand side of ), in turn, attains its maximum at n E C if E C n )α α if E C > n )α The result is obtained by substituting this value of γ into ) and recalling the known fact that E C α see, eg, [9, pp 55 56], where this fact is used in proving the Johnson bound) We next compare Theorem 5 with the DLP bound for constant-weight codes counterpart of 1) for the constant-weight case is given by where B l = 1 M l The w E l i; n, w) B i, 0 l w, 3) ) ) ) w x n w x E l x; n, w) = j=0 x 1) j j l j l j ) ) w n w, 0 l w, l l 16

18 and the constraints 13) now become B l 0, 0 l w ) see [1, Proposition 16] and [9, p 665]) In particular, for l = we get ) ) n x αn, w) 1) x E x; n, w) = ) ) + 1 w n w and, so, ) ) w n w 0 16 B ) ) ) ) n w n w = S C α 1)E C + 16 ) = ) n )S C nα 1)E C + n α nα Hence, S C 1 ) nα 1)E C n α n + nα 5) A simple check reveals that the bound of Theorem 5 is stronger than 5) whenever E C < nα 1) n nα For larger values of E C, the bound 5) becomes tighter, until the bounds coincide again at E C = α When w = n/, the value of α is n/, in which case the bound 5) coincides with the bound of Theorem 5 whenever E C is in the range nn ) ) = n )α E C α = n ; this range is precisely where the bound of Theorem 5 is linear in E C Example 51 We consider the parameters n = 8 and d max /d = /3 By Theorem 5, the largest nearly-equidistant n=8, M, d=6, w) code has size M 63, and equality is possible only when w = For this value of w, the largest possible size M is attained when E C = α8, ) = 8/7, in which case Theorem 5 coincides with 5); therefore, for w =, Theorem 5 does not offer an improvement over the constant-weight DLP bound However, for larger values of w, Theorem 5 strictly improves on that DLP bound For example, 17

19 for w = 1, Equations 3) ) allow M = 80 as a feasible solution, while Theorem 5 bounds M from above by 58 Similarly, by combining Equations 3) ) with Theorem 5, we get that the largest n=8, M, d=1, w) code with d max = 16 has size M 63, and equality is possible only when w {11, 1} Equality is indeed attained by codes with w = 1 that are based on quasi-symmetric -designs [10], [11, p 163] Table 3 is the constant-weight counterpart of Table for code size M = 6) The first Range of n d max /d n = 8 9 n n 1 15 n 18 5/3 19 n 3 3/ n 8 7/5 9 n 3 /3 Table 3: Lower bounds on the ratio d max /d for n, M=6) constant-weight codes, as a function of n five rows in Table 3 present tight bounds Ratios of and 3, respectively, can be achieved by the trivial) 8, 70,, ) and 9, 8,, 3) codes A ratio of d max /d = is attained by the 11, 66,, 5) code that is based on the Steiner system S, 5, 11) [9, p 70], and a ratio of d max /d = 5/3 is attained by the 15, 70, 6, 6) code that consists of the codewords of Hamming weight 6 in the shortened 15, 18, 6) Nordstrom Robinson code [9, p 7] Finally, one can obtain a 19, 70, 8, 9) code with d max /d = 3/ by shortening, expurgating, and augmenting the set of codewords of Hamming weight 1 in the extended binary Golay code see the Appendix in [8]) Figure depicts the bounds of Theorem 5 and Equation 5), superimposed on Figure 1 Figure is drawn to scale for n = 5 and w = ) References [1] E Agrell, A Vardy, K Zeger, Upper bounds for constant-weight codes, IEEE Trans Inform Theory, 6 000), [] A Ashikhmin, A Barg, Binomial moments of the distance distribution: bounds and applications, IEEE Trans Inform Theory, ),

20 S C α, n α ) n, nn+1) ), n) ) Theorem 33 Equation 11) Theorem 5 E C E C Equation 5) 1 Figure : Bounds for the constant-weight case E C [3] A Ashikhmin, A Barg, S Litsyn, Estimates of the distance distribution of codes and designs, IEEE Trans Inform Theory, 7 001), [] T Beth, D Jungnickel, H Lenz, Design Theory, Cambridge University Press, Cambridge, 1986 [5] A Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combinatoria, ), [6] AE Brouwer, JB Shearer, NJA Sloane, WD Smith, A new table of constant weight codes, IEEE Trans Inform Theory, ), [7] LD Grey, Some bounds for error-correcting codes, IRE Trans Inform Theory, 8 196), 00 0 and 355 [8] PJ Kuekes, W Robinett, RM Roth, G Seroussi, GS Snider, RS Williams, Resistor-logic demultiplexers for nanoelectronics based on constant-weight codes, Nanotechnology, ), [9] FJ MacWilliams, NJA Sloane, The Theory of Error-Correcting Codes, North- Holland, Amsterdam,

21 [10] C Parker, E Spence, VD Tonchev, Designs with the symmetric difference property on 6 points and their groups, J Comb Theory A, ), 3 3 [11] VD Tonchev, Codes and designs, in Handbook of Coding Theory, VS Pless, WC Huffman Editors), Elsevier, Amsterdam, 1998, pp [1] JH Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford,