Des. Codes Cryptogr. (2011) 60:81–89DOI 10.1007/s10623-010-9418-4

On binary linear r-identifying codes

Sanna Ranto

Received: 29 January 2010 / Revised: 1 July 2010 / Accepted: 5 July 2010 /Published online: 20 July 2010© Springer Science+Business Media, LLC 2010

Abstract A subspace C of the binary Hamming space Fn of length n is called a linearr -identifying code if for all vectors of Fn the intersections of C and closed r -radius neighbour-hoods are nonempty and different. In this paper, we give lower bounds for such linear codes.For radius r = 2, we give some general constructions. We give many (optimal) constructionswhich were found by a computer search. New constructions improve some previously knownupper bounds for r -identifying codes in the case where linearity is not assumed.

Keywords Identifying codes · Hamming space · Linear codes

Mathematics Subject Classification (2000) 05C70 · 94B05 · 94C12

1 Introduction

Let F = {0, 1} and the n-fold Cartesian product of F is the binary Hamming space of dimen-sion n, it is denoted by Fn . We call the elements of Fn words or vectors. We denote by 0 the allzero word and by 1 the all one word. Let x and y be vectors in Fn . The (Hamming) distance,d(x, y), between x and y is the number of coordinate places where they differ. The weight ofx is w(x) = d(x, 0). The support of x, supp(x), is the set of the nonzero coordinates of x. Anonempty subset C of the words of Fn is called a code and its elements are called codewords,n is the length of a code. We assume that a code contains always at least two codewords.The minimum distance of a code C is

d(C) = minc1,c2∈C

d(c1, c2).

A code C is called linear if it is a subspace of Fn .

Communicated by J. D. Key.

S. Ranto (B)Department of Mathematics, University of Turku, 20014 Turku, Finlande-mail: samano@utu.fi

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82 S. Ranto

We denoteBr (x) = {y ∈ Fn | d(x, y) ≤ r},Sr (x) = {y ∈ Fn | d(x, y) = r},

where Br (x) is the Hamming sphere of radius r centered at x. The cardinality of Hammingsphere of radius r and length n is denoted by V (n, r), the cardinality does not depend on thecenter.

Moreover, we say that x ∈ Fn is r -covered by y ∈ Fn if y ∈ Br (x). Let C ⊆ Fn be a code,then we denote

Ir (C; x) = Ir (x) = Br (x) ∩ C.

If |Ir (C; x)| = μ, then we say that x is μ-fold r -covered.For x = (x1, . . . , xn) and y = (y1, . . . , yn) in Fn we have

x + y = (x1 + y1, . . . , xn + yn)

and for a set A⊆Fn we have x + A = {x + a | a ∈ A}.Definition 1 A code C ⊆ Fn is called an r -covering if Ir (C; x) �= ∅ for all x ∈ Fn .

Definition 2 A code C ⊆ Fn is r -identifying (or an r -identifying code) if for all x in Fn thesets Ir (C; x) are nonempty and different.

Naturally, we are interested in the smallest possible cardinality of an identifying code fordifferent lengths. Any code attaining the smallest possible cardinality is called optimal. Thecardinality of an optimal r -identifying code of length n is denoted by Mr (n). The value ofMr (n) is considered in several papers, see for example [1–8,12].

In this paper, we are interested in binary linear r -identifying codes, meaning that anr -identifying code is a subspace of Fn . The smallest possible dimension of a linear r -identifying code in Fn is denoted by kr [n]. Clearly, Mr (n) ≤ 2kr [n]. A code attaining thesmallest possible dimension is called optimal. Previously, binary linear 1-identifying codeshave been considered in [13], and in [7] it has been proven that the problem whether a givenbinary linear code is r -identifying is �2-complete.

Identifying codes were introduced by Karpovsky et al. [8]. The motivation for these codeswas finding malfunctioning processors in multiprocessor systems. Now identifying codesconstitute a topic of their own with over 150 papers written about them and closely relatedareas, see [10]. The motivation of identifying codes can also be found for example fromsensor networks, see [9].

The structure of the paper is the following: In Sect. 2 we give two lower bounds for thedimension of an r -identifying code. In Sect. 3 we give constructions for linear 2-identifyingcodes, these constructions give us longer codes from shorter ones. In Sect. 4 in Table 2 weenumerate the parity check matrices of linear r -identifying codes for 2 ≤ r ≤ 5 whichwe found by a computer search. We also give some improvements on the cardinalities ofr -identifying codes.

2 Lower bounds

In [13] Lemmata 3 and 4 and Theorem 5 have been proven for radius r = 1. The proofs forlarger r are similar to the one of r = 1.

Lemma 3 Let C ⊆ Fn be a linear code. For all x ∈ Fn and for all c ∈ C, we haveIr (x + c) = Ir (x) + c.

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On binary linear r -identifying codes 83

Proof y ∈ Ir (x + c) ⇔ y ∈ C and d(x + c, y) ≤ r ⇔ y + c ∈ C and d(x, y + c) ≤ r ⇔y + c ∈ Ir (x) ⇔ y ∈ Ir (x) + c. Lemma 4 Let C ⊆ Fn be a linear r-identifying code. Then for all x ∈ Fn |Ir (C; x)| �= 2.

Proof Let y ∈ Fn . Suppose to the contrary that |Ir (y)| = 2, then Ir (y) = {y + a1, y + a2}for some a1, a2 ∈ Fn(a1 �= a2) where w(ai ) ≤ r(i = 1, 2). Now {y + a1, y + a2} ⊆ Ir (y +a1 + a2). Since C is linear a1 + a2 ∈ C and by Lemma 3 we get |Ir (y)| = |Ir (y + a1 + a2)|.This is a contradiction.

Lemma 4 implies that if C is a linear r -identifying code, then every word in Fn isr -covered either by exactly one codeword or by at least three codewords. Hence, the fol-lowing lower bound follows by double counting the number of ordered pairs (x, c) wherex ∈ Fn and c ∈ C.

Theorem 5

kr [n] ≥⌈

log23 · 2n∑r

i=0

(ni

) + 2

⌉.

The following lemma is from [6].

Lemma 6 ([6]) Suppose that r < n − r − 1. If a code C ⊆ Fn is r-identifying, then it is(n − r − 1)-identifying. If a code C ⊆ Fn is (n − r − 1)-identifying, then it is r-identifyingunless there exists a unique word x ∈ Fn such that Ir (C; x) = ∅.

Proof Suppose first that C is an r -identifying code. Then for all x ∈ Fn we haveIn−r−1(C; x) = C\Ir (C; x + 1). Sets C\Ir (C; x + 1) are all different since C isr -identifying. These sets are also nonempty because otherwise if Ir (C; x + 1) = C thenbecause r < n − r − 1 we have Ir (C; x) = ∅, which is impossible.

Now the proof of the second statement goes similarly. Theorem 7 kr [n] = kn−r−1[n].Proof Without loss of generality we may assume r < n − r − 1. Let C ⊆ Fn be a linearr -identifying code. Then by Lemma 6 we know that C is (n−r −1)-identifying. Assume thenthat C⊆Fn is a linear (n −r −1)-identifying code. By Lemma 6 we know that C⊆Fn is a lin-ear r -identifying code, unless there exists a unique word x ∈ Fn such that C\In−r−1(x) = ∅.

Suppose to the contrary that such x would exist. Because C is linear, for every c ∈ C we haveby Lemma 3 that C\In−r−1(x + c) = ∅. Hence, In−r−1(x) = In−r−1(x + c) = C, which isimpossible.

By Lemma 4 we can improve the lower bound from [8] and [4] for linear codes, see theproof in [4]. We get the bound in Theorem 8 which improves some values compared withTheorem 5. In most cases, these two bounds coincide.

Theorem 8 Let C ⊆ Fn be a linear r-identifying code. Then

|C | · V (n, r) ≥ |C | +s∑

i=3

i

( |C |i

)+ (s + 1)

(2n − |C | −

s∑i=3

( |C |i

))(1)

where s is the largest integer such that

|C | +s∑

i=3

( |C |i

)≤ 2n . (2)

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84 S. Ranto

When n/2 ≤ r ≤ n − 1, then instead of (1) we use

|C | · V (n, n − r − 1) ≥ |C | +s∑

i=3

i

( |C |i

)+ (s + 1)

(2n − |C | −

s∑i=3

( |C |i

)). (3)

When applying the lower bound of Theorem 8 for an r -identifying code of length n wefirst choose some (small) |C |. Then we count the largest integer s which satisfies (2).We continue by trying whether these |C | and s satisfy (1) (or (3) depending on r ), if notwe increase |C | by one and start the process again. We continue until we find a suitable|C |. When n/2 ≤ r ≤ n − 1, we use (3) because then V (n, n − r − 1) ≤ V (n, r) and byTheorem 7 we know that a linear (n − r − 1)-identifying code is also r -identifying.

3 Linear 2-identifying codes

Theorem 9 Let C ⊆ Fn be a linear code with d(C) = 3. The code C is 2-identifying if andonly if every noncodeword is at least 3-fold 2-covered.

Proof Suppose first that C is 2-identifying, then because d(C) = 3 every codeword is2-covered only by itself, and Lemma 4 implies that every noncodeword is at least 3-fold2-covered.

Suppose then that every noncodeword is at least 3-fold 2-covered. By the assumptiond(C) = 3, every codeword is distinguishable from all the other words.

Let x ∈ Fn\C. Because d(C) = 3, there can be at most one codeword at distance onefrom x. Suppose first that there is c1 ∈ S1(x) ∩ C , then there are c2, c3 ∈ S2(x) ∩ Csuch that d(c2, c3) = 4 and d(c1, c2) = d(c1, c3) = 3. Let us denote supp{x + c1} ={i1}, supp{x + c2} = {i2, i3} and supp{x + c3} = {i4, i5}, where i j �= ih when j �= h.

Without loss of generality we may assume that i j = j for j = 1, . . . , 5. Hence, the questionis whether there is a word y �= x such that d(y, x + 10 . . . 0) ≤ 2, d(y, x + 0110 . . . 0) ≤ 2and d(y, x + 000110 . . . 0) ≤ 2. It is easy to see that such y does not exist. Hence, there doesnot exist y �= x such that {c1, c2, c3}⊆I2(y).

If there does not exist a codeword at distance one from x, then there are at least threecodewords in S2(x) whose mutual distances are four. Without loss of generality we mayassume that c1 = x + 110 . . . 0, c2 = x + 00110 . . . 0 and c3 = x + 0000110 . . . 0 belong toI2(x). The question is now whether there exists y �= x such that d(y, c1) ≤ 2, d(y, c2) ≤ 2and d(y, c3) ≤ 2. Again, it is easy to see that such y does not exist.

Theorem 9 gives a way to construct linear 2-identifying codes. Let C ⊆ Fn be a linearcode of dimension k and let H be a (n −k)×n parity check matrix of C. This means c ∈ C ifand only if HcT = 0T (here xT means the transpose of a vector x). Now, every noncodewordin Fn\C is at least 3-fold 2-covered if for every y ∈ Fn−k\{0} there are at least three differentvectors x1, x2, x3 ∈ Fn of weight at most 2 such that HxT

i = yT for i = 1, 2, 3. Moreover,the minimum distance of C is three if 0T is not a column of H , there are not two samecolumns in H and there are some three columns with sum equal to 0T . For the results on(parity) check matrices see e.g. [11].

By Theorem 9 and above notions it is clear that the following constructions in Theo-rems 10, 11, and 12 yield linear 2-identifying codes.

Theorem 10 Let C ⊆ Fn be a linear 2-identifying code with d(C) = 3 and a dimensionk. Let H = (h1 | . . . | hn) (where hi are column vectors) be a parity check matrix of C.

If y ∈ Fn−k\{0T , h1, . . . , hn}, then

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On binary linear r -identifying codes 85

H ′ = (y | h1 | . . . | hn)

is a parity check matrix for a linear 2-identifying code of length n + 1 and dimension k + 1.

Theorem 11 Let C ⊆ Fn be a linear 2-identifying code with d(C) = 3 and a dimension k.

Let H be the parity check matrix of C. Then

H ′ =(

1 0 1H H 0T

)

is a parity check matrix of a linear 2-identifying code of length 2n + 1 and dimension n + k.

Theorem 12 Let C1 ⊆ Fn1 and C2 ⊆ Fn2 be linear 2-identifying codes where d(Ci ) = 3and dimensions dim(Ci ) = ki for i = 1, 2. Let

Hi = (h(i)1 | . . . | h(i)

ni)

where h(i)j (for j = 1, . . . , ni ) are column vectors of length ni − ki , be a parity check matrix

of Ci for i = 1, 2. Now

H =(

h(1)1 . . . h(1)

1H2

∣∣∣∣ . . .. . .

∣∣∣∣ h(1)n1 . . . h(1)

n1

H2

∣∣∣∣ H1

0

∣∣∣∣ 0H2

)

is a parity check matrix of a linear 2-identifying code of length n = n1 + n2 + n1n2 anddimension k = n1n2 + k1 + k2.

Example 13 If we choose in Theorem 12 C1 = C2 to be a linear 2-identifying code of lengthsix and dimension 3 (the check matrix is in Sect. 4), then we receive a linear code of length48 and dimension 42. By Theorem 5 k2[48] ≥ 40.

Let C1 ⊆ Fn and C2 ⊆ Fm . The direct sum of codes C1 and C2 is

C1 ⊕ C2 = {(c1, c2) | c1 ∈ C1, c2 ∈ C2} ⊆ Fn+m .

If C1 and C2 are linear codes, then clearly C1 ⊕ C2 is also a linear code. Moreover, if C1 isan r -covering and C2 is an s-covering then C1 ⊕ C2 is (r + s)-covering.

Remark 14 By [13] we know that for every n ≥ 3 there exists an optimal linear 1-identi-fying code C ⊆ Fn such that d(C) = 2 and for every x ∈ Fn\C we have |I1(C; x)| ≥ 3.

This implies that for all x, y ∈ Fn\C, x �= y, there exist c ∈ I1(C; x)\I1(C; y) and c′ ∈I1(C; y)\I1(C; x).

In [5] it has been proven that M2(n + m) ≤ M1(n)M1(m). Now we prove a similar resultfor linear identifying codes.

Theorem 15 If n ≥ 5 and m ≥ 5, then k2[n + m] ≤ k1[n] + k1[m].Proof Let C1 ⊆ Fn and C2 ⊆ Fm be optimal linear 1-identifying codes. By Remark 14, C1

and C2 can be chosen in such a way that (for i = 1, 2)d(Ci ) = 2 and for every x ∈ Fn\Ci

we have |I1(Ci ; x)| ≥ 3. We will show that the direct sum C1 ⊕ C2 ⊆ Fn+m is a linear2-identifying code.

Denote D = C1 ⊕ C2. Let x = (x1, x2), y = (y1, y2) ∈ Fn+m , where x1, y1 ∈ Fn andx2, y2 ∈ Fm . We will show that

(I2(D; x)\I2(D; y)) ∪ (I2(D; y)\I2(D; x)) �= ∅for all x �= y. Without loss of generality we may assume that x1 �= y1.

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86 S. Ranto

(1) Suppose first that x1 �∈ C1 and y1 �∈ C1. By Remark 14 we have c1 ∈I1(C1; x1)\I1(C1; y1). If x2 �= y2 or x2 = y2 �∈ C2 there is c2 ∈ I1(C2; x2) suchthat d(c2, y2) ≥ 1. Then (c1, c2) ∈ I2(D; x)\I2(D; y).

Suppose then x2 = y2 ∈ C2. If d(c1, y1) ≥ 3, we are done because (c1, x2) ∈I2(D; x)\I2(D; y). If d(c1, y1) ≤ 2 for all c1 ∈ I1(C1; x1), it follows that d(x1, y1) =1. (Namely, the cases 2 ≤ d(y1, x1) ≤ 3 are impossible. If d(y1, x1) = 2, then|B1(y1)∩B1(x1)| = 2, and there always exists c0 ∈ I1(C1, x1) such that d(c0, y1) = 3.For the case d(y1, x1) = 3 we notice that by Remark 14 we have three distinct vectorse1, e2 and e3 of weight one such that x1+e1, x1+e2 and x1+e3 are in I1(C1; x1)∩S2(y1).

Because d(x1, y1) = 3 the sum x + e1 + e2 + e3 is necessarily equal to y1, and bylinearity, belongs to C1, which is impossible.) Because y1 �∈ C1, there are at least threecodewords that 1-cover it, and consequently, y1 cannot be the only noncodeword atdistance one from x1 : otherwise d(C1) = 2 would not hold. Take any such z, then byRemark 14 there is cz ∈ I1(C1; z) such that d(cz, x1) = 2 and d(cz, y1) = 3. Now(cz, x2) ∈ I2(D; x)\I2(D; y).

(2) Suppose then x1 ∈ C1 and y1 �∈ C1 (the case y1 ∈ C1 and x1 �∈ C1 can be doneanalogously). Now there is c1 ∈ I1(C1; y1) such that d(c1, x1) ≥ 2. If x2 �= y2 orx2 = y2 �∈ C2 there is c2 ∈ I1(C2; y2) such that d(c2, x2) ≥ 1. Then (c1, c2) ∈I2(D; y)\I2(D; x).

Suppose then x2 = y2 ∈ C2. Because d(C2) = 2 every z such that d(z, x1) = 1satisfies z �∈ C2, and therefore there exists c2 ∈ C2 such that d(c2, x2) = 2. Becaused(x1, y1) ≥ 1, we have (x1, c2) ∈ I2(D; x)\I2(D; y).

(3) Suppose finally x1 ∈ C1 and y1 ∈ C1 (now d(x1, y1) ≥ 2). If there is c2 ∈ I1(C2; x2)

such that d(c2, y2) ≥ 1, then (x1, c2) ∈ I2(D; x)\I2(D; y). If such a codeword c2 doesnot exist, then x2 = y2 ∈ C2. If this is the case, there is c2 ∈ C2 such that d(c2, x2) = 2.

Hence, (x1, c2) ∈ I2(D; x)\I2(D; y).

4 Computer results

4.1 The computer search of codes

Two linear codes C1 and C2 are equivalent if C2 can be obtained by applying fixed permu-tations to the columns of C1.

It is well known that all the nonequivalent linear codes of length n and dimension k canbe represented as (n − k) × n parity check matrices in standard form (A | I ) where A is a(n − k) × k matrix and I is the (n − k)-dimensional identity matrix. Hence, when we searchparity check matrices for linear r -identifying codes we only need to consider matrices instandard form. Moreover, Lemma 4 indicates that all, except one, vectors of length n − khave to be reached at least three different times as the sum of at most r column vectors ofthe suitable check matrix. If there is an exceptional one column, then it has to be the sumof at most r columns exactly in one way (including the empty sum). After we have founda matrix which satisfies the earlier properties we only need to check whether all the wordsin the Hamming sphere of radius r centered at the all zero word are all distinguishable fromeach other using codewords of weight at most 2r. Namely, because a code is linear the neigh-bourhood of every codeword is similar by Lemma 3. Hence, for two words x and y we canhave Ir (x) = Ir (y) if and only if they belong to the same r -radius sphere of some codeword.

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On binary linear r -identifying codes 87

Table 1 Bounds on thedimensions of linearr -identifying codes

The lower bounds for kr [n] for2 ≤ r ≤ 5 come from Theorem 5unless stated something else* An improvement to thepreviously known cardinality ofan r -identifying code. The formerrecords are in [3].a Theorem 8e Exhaustive computer searchA Theorem 7B A construction in Sect. 4C Theorem 10D By [5]:kr [n + r + i] ≤ kr [n] + r + i,for i ≥ 1.

E Theorem 11

n k2[n] k3[n] k4[n] k5[n]2 – – – –

3 3 – – –

4 3 A 4 – –

5 e 4 4 A 5 –

6 3 B 3 A 5 A 6

7 4 C 3 B 4 A 6 A

8 5 B 4 B 4 A 5 A

9 5 B 4* B a 4 B 4* A

10 6 C 5 B a 4 B a 4 A

11 7 C e 6 B e 5 B e 5 B

12 8 C 6* B a 5* B e 5 B

13 9 C, E 7 B e 6 B a 5 B

14 e 10 C 8 B 6* B a 5–6 B

15 10 B 8* B 6–7 B 5–6 B

16 11 C 9* B 7–8 B a 6–7 B

17 12 C 9–10 B 7–8 B 6–7* B

18 13 C 10–11 B 8–9 B a 7–8 B

19 13–14 C, E 11–12 D 9–10 B 7–8* B

20 14* B 12* B 9–10* B 8–9* B

21 15* C 12–13* B 10–11* B 8–10* B

22 16 C 13–14 B

23 17 C 14–15 B

24 18 C 15–16 B

25 19 C 16–17 D

26 20 C 17–18 D

27 21 C 17–18 B

28 21–22 C 18–19 B

29 22–23 C 19–20 B

30 23–24 C 20–22 D

In Table 1 we summarize the known bounds for the dimensions of linear r -identifyingcodes for 2 ≤ r ≤ 3 for lengths r + 1 ≤ n ≤ 30 and 4 ≤ r ≤ 5 for lengths r + 1 ≤ n ≤ 21.

By [1] we know that kr [r + 1] = r + 1.

In Table 2 we enumerate all the check matrices which we found by a computer search.

4.2 Some code improvements

From some linear r -identifying codes, which have a minimum distance less than r + 1, wecan remove a codeword and the remaining code is still r -identifying (not linear any more).If we remove one codeword from a linear code, it does not matter which one we choose tobe removed, the effect is the same. By removing the all zero word from linear r -identifyingcodes mentioned in Table 2 we obtained the improvements on the cardinalities of r -identify-ing codes in the following theorem. From codes of length 15 and radii 3 and 5 we also could

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88 S. Ranto

Table 2 The columns of matrices A, when (A | In−k ) is a parity check matrix of a linear r -identifying codeof length n and dimension k

r n k The columns of A

2 6 3 6 5 3

2 8 5 0 4 2 5 3

2 9 5 3 5 10 12 15

2 15 10 3 5 6 7 9 10 11 20 24 28

2 20 14 45 54 24 61 46 15 27 42 49 51 13 26 14 9

3 7 3 3 5 6

3 8 4 3 5 6 7

3 9 4 3 12 21 26

3 10 5 3 3 12 21 26

3 11 6 30 6 27 23 15 17

3 12 6 0 3 13 22 39 56 32

3 13 7 3 12 21 26 38 43 50

3 14 8 25 62 48 11 24 2 52 22

3 15 8 59 66 102 41 119 90 43 79

3 16 9 8 125 86 100 91 54 19 80 42

3 17 10 88 17 102 115 35 122 52 116 54 91

3 18 11 63 12 79 93 17 125 84 82 23 4 68

3 20 12 99 166 159 137 113 237 24 214 210 59 69 139

3 21 13 16 254 122 134 111 97 247 21 210 198 225 76 58

3 22 14 131 57 243 202 236 202 80 191 151 210 161 251 141 178

3 23 15 82 66 252 150 75 204 82 164 223 2 174 124 187 153 15

3 24 16 56 178 251 227 97 6 215 213 239 45 2 19 132 198 15 143

3 27 18 69 422 503 41 81 268 147 484 257 255 241 306 106 50 201 92 362 373

3 28 19 318 425 257 233 452 112 183 142 134 71 327 182 42 285 375 434 361 81 315

3 29 20 283 202 498 392 14 295 434 6 149 48 85 197 497 508 195 328 484 409 23 430

4 9 4 3 5 10 12

4 10 4 3 12 21 26

4 11 5 29 3 6 25 21

4 12 5 57 79 24 91 47

4 13 6 127 98 93 72 6 26

4 14 6 59 101 247 231 5 191

4 15 7 74 67 129 217 78 254 119

4 16 8 112 224 137 117 51 63 156 131

4 17 8 60 172 147 451 393 378 205 109

4 18 9 175 340 125 239 407 180 399 242 225

4 19 10 178 443 56 10 233 202 340 389 185 103

4 20 10 954 571 342 1013 115 642 874 542 93 994

4 21 11 520 615 423 425 37 122 885 283 34 654 197

5 11 5 39 51 2 19 20

5 12 5 0 3 28 45 54

5 13 5 210 31 172 119 85

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On binary linear r -identifying codes 89

Table 2 continued

r n k The columns of A

5 14 6 254 201 211 15 52 135

5 15 6 301 377 227 438 297 225

5 16 7 133 436 432 335 105 368 502

5 17 7 280 489 948 269 898 226 46

5 18 8 155 971 384 385 339 63 945 484

5 19 8 911 807 1762 539 1445 474 1214 98

5 20 9 1958 1325 1956 629 1895 1736 172 1009 243

5 21 10 1023 1210 854 339 2025 183 1235 532 1095 1681

The columns are the binary representation of the integers

remove an other word (the word with highest weight). The previous records (in parentheses)are from [3].

Theorem 16 M3(15) ≤ 254(305), M4(14) ≤ 63(76), M5(9) ≤ 150 (17), M5(10) ≤15(16), M5(15) ≤ 62(64).

Acknowledgements The author wishes to thank the anonymous reviewers for numerous suggestions thatgreatly improved the quality of the paper.

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