DIMs of Odd Length

We cannot construct a binary n-DIM for odd n in the same way as Construction 1 because it is infeasible to find two commutative sets which form a basic construction set when n is odd. In the following, we develop a different construction method for odd n.

Lemma 3.3 Let n = 2m + 1, m ≥ 2, fn be a mapping constructed by (3.1) with the

Chapter 3 DIMs from Binary Vectors 24

following basic construction set

fn

The following shows that (3.8) is true in all possible cases.

Chapter 3 DIMs from Binary Vectors 25 union of any k distinct element of V. We have

) statements are true.

i) d ( _{\{ 1}} , _{m 1}) n.

Proof. First, i) implies that (3.3) in Lemma 3.1 is true. Second, for any two distinct subsets J1, J2⊆ Fn, there are three possible cases:

1. Neither J1 nor J2 contains m + 1.

2. Either J1 or J2 contains m + 1 but not both.

3. Both J1 and J2 contain m + 1.

No matter in which case, we show that (3.2) in Lemma 3.1 is always true.

Case 1: m + 1 ∉ J1 and m + 1 ∉ J2. This case is basically the same situation as in Theorem 3.1 above. Thus

)

Chapter 3 DIMs from Binary Vectors 26

Case 2: Without loss of generality, assume m + 1 ∈ J1 and m + 1 ∉ J2. We prove (3.2) by induction on the size of J1 ⊕ J2. The base step is stated in ii) for

| J1⊕ J2| = n – 1. Now assume (3.2) is true for | J1 ⊕ J2 | = k + 1 but is not true for | J1⊕ J2| = k. That is,

) , (

∏

j∈J1⊕J2 j ι

dH ρ ≤ k

for some | J1⊕ J2| = k. However, the only possibility for this assumption is )

, (

∏

j∈J1⊕J2 j ι

dH ρ = k.

Because according to the hypothesis,

) ,

( _{}

2

1 ι

∏

j∈J ⊕J i j

dH

U ρ > k + 1

for all i∈ Fn – (J1 ⊕ J2), and ρi is a transposition that changes exactly two positions (note that m + 1 ∈ J1 ⊕ J2). Thus,

∏

j∈J₁⊕J₂ρj agrees with ι in n – k positions, and each permutation ρ_i such that i ∈ F_n – (J1 ⊕ J2) changes exactly two of these positions to make

) ,

( _{}

2

1 ι

∏

j∈J ⊕J i j

dH

U ρ = k + 2.

There are totally n – k permutations each corresponding to an element of F_n– (J1⊕ J2). By the same logic as in Lemma 3.3, it is not possible for those n – k permutations, which consist of two commutative sets and one of them is of size≥

⎡ ⎤

n−2k , that change only n – k positions, which is a contradiction! Thus, we have

) , (

∏

j∈J1⊕J2 j ι

dH ρ > k for | J1 ⊕ J2 | = k.

Case 3: m + 1 ∈ J1 and m + 1 ∈ J2. According to Lemma 3.3, we have

Chapter 3 DIMs from Binary Vectors 27

) ,

(

∏

j∈J1 j

∏

j∈J2 j

dH ρ ρ =

∏

∈ ⊕ _{ +_1} +₁

2

1 , )

( j J J m j m

dH

U ρ ρ

> | J1 ⊕ J2 |.

So if we can find a ρm+1 satisfying i), ii), and iii) in Lemma 3.4, then we have a binary n-DIM for odd n. The following examples exhibit how to find a suitable permutation for ρm+1 for n = 5 and n = 7 respectively.

Example 3.3 (n = 5) Suppose f5 : Z₂⁵→ S5 is constructed by (3.1) with the following basic construction set

f5

B = 〈 ρ1 = (2, 1, 3, 4, 5), ρ2 = (1, 2, 4, 3, 5), ρ3 = (π1, π2, π3, π4, π5), ρ4 = (1, 3, 2, 4, 5), ρ5 = (1, 2, 3, 5, 4) 〉.

To make f5 a DIM, the following requirements should be satisfied:

i) dH (ρ1ρ2ρ4ρ5, ρ3) = 5.

ii) d_H (ρ2ρ4ρ5, ρ3) = 5, d_H (ρ1ρ4ρ5, ρ3) = 5, d_H (ρ1ρ2ρ4, ρ3) = 5, and d_H (ρ1ρ2ρ5, ρ3) = 5.

iii) {π2, π3}, {π4, π5}∉ {{1, 2}, {3, 4}} and {π2, π3, π4, π5} ≠ {1, 2, 3, 4}.

Since

ρ1ρ2ρ4ρ5 = (2, 4, 1, 5, 3), ρ2ρ4ρ5 = (1, 4, 2, 5, 3), ρ1ρ4ρ5 = (2, 3, 1, 5, 4), ρ1ρ2ρ5 = (2, 1, 4, 5, 3), ρ1ρ2ρ4 = (2, 4, 1, 3, 5),

we have π1 ∉ {1, 2}, π2 ∉ {1, 3, 4}, π3 ∉ {1, 2, 4}, π4 ∉{3, 5}, and π5 ∉ {3, 4, 5}.

Furthermore, from iii) above we have π1 ≠ 5. According to these restrictions and the

Chapter 3 DIMs from Binary Vectors 28

rules stated in iii), the only solution for ρ3 is (3, 2, 5, 4, 1). The mapping table of f5 is listed in Appendix A and the distance expansion matrix of f5 is listed in Table 3.6.

Example 3.4 (n = 7): Assume f7 : Z₂⁷→ S7 is constructed by (3.1) with the basic construction set described in Lemma 3.3. Based on the requirements depicted in Lemma 3.4, we exclude some values for ρ4 in the same way as Example 3.1. The excluded values are summarized in Table 3.1.

Table 3.1 The excluded values for ρ4 in the construction of f7 ∈ I (2, 7, 7).

π1 π2 π3 π4 π5 π6 π7

1 × × ×

2 × ×

3 × × ×

4 × × ×

5 × × ×

6 × × ×

7 × × ×

In Table 3.1 the marks “×” denote the values that should be excluded. Besides, the selection of the values should satisfy the condition iii) in Lemma 3.4. There are many solutions for ρ4 (totally 68). In order to make the distance expansion matrix as good as possible, we can choose a solution such that dH (ρ4, ι) is the largest among all possible solutions, for example, (5, 6, 3, 7, 1, 2, 4). The mapping table of f7 is listed in Appendix A and the distance expansion matrix of f7 is listed in Table 3.12.

Now we give a general construction of binary n-DIMs for odd n as follows.

Construction 3.2 Let n = 2m + 1 and m ≥ 2. Construct a mapping fn with the basic construction set described in Lemma 3.3 where

Chapter 3 DIMs from Binary Vectors 29

Theorem 3.2 The mapping fn constructed by Construction 3.2 is a DIM for odd n.

Proof. We have shown that f5 and f7 are DIMs from the above examples. For n ≥ 9, summarized in Table 3.2 where the marks “×” denote the values excluded and the marks “○” denote the values selected. It can be checked that ρ_m+1 satisfies iii) in Lemma 3.4.

Chapter 3 DIMs from Binary Vectors 30

3.3 Comparisons

In this section, we compare our DIMs fn with other mappings, including DPMs hn proposed by Chang et al. [17], DPMs ln of odd length proposed by Lee [23], DIMs rn proposed by Chang [14], DIMs Qn proposed by Chang [15], DIMs zn proposed by Lee [22], and DPMs Mn proposed by Lee [31]. Tables 3.3 ~ 3.27 list the distance expansion matrices of these mappings for comparisons for 5 ≤ n ≤ 11. Table 3.28 lists the distance expansion matrix of fn for n = 13. The total distances ∆1( f ) of these mappings are listed in Table 3.29. The asterisk behind a number indicates that this number is the largest among all items. In the comparisons of distance expansion distributions, we only compare fn with hn, ln, and rn for 5 ≤ n ≤ 9, and also compare fn

with Qn for 8≤ n ≤ 10 because only those mappings are given in the above-mentioned papers. We do not compare fn with Mn for their distance expansion matrices since there is no such matrix in the corresponding paper. For even n, the distance expansion distribution of zn and fnare exactly the same since zn and fn are identical when n is even.

For n = 5, Tables 3.3 and 3.4 show that both h5 and l5 are DPMs but not DIMs, whereas Tables 3.5 and 3.6 show that r5, z5, and f5 are all DIMs (the distance expansion distribution of z5 and f5 are exactly the same). The distance expansion distribution of r5 is better than that of z5 and f5, and the total distances of these mappings justify this argument. This is reasonable since r5 is obtained by computer search.

Table 3.3 Distance expansion matrix of h5.

0 80 0 0 0

0 96 64 0

0 112 48 16 64 16

Chapter 3 DIMs from Binary Vectors 31

Table 3.4 Distance expansion matrix of l5.

0 64 6 2 8

4 68 64 24

14 76 70 22 58 16

Table 3.5 Distance expansion matrix of r5.

0 49 8 10 13

0 68 68 24 0 93 67 0 80 16

Table 3.6 Distance expansion matrix of z5 and f5.

0 64 16 0 0

0 48 112 0

0 64 96

0 80 16

For n = 6, l6 is not compared since the paper [23] focuses on DPMs of odd length only. Although f6 is not identical to h6 and r6 (h6 = r6), the distance expansion matrices of these mapping is just the same (see Table 3.7).

Table 3.7 Distance expansion matrix of h6, r6, z6, and f6.

0 192 0 0 0 0

0 192 288 0 0 0 192 384 64 0 192 288

0 192 32

Chapter 3 DIMs from Binary Vectors 32

For n = 7, we see again that both h7 and l7 are DPMs but not DIMs, whereas r7, z7, and f7 are all DIMs (see Tables 3.8 ~ 3.12). One notable thing is that the distance expansion distribution of f7 is better than that of r7 and z7, and the total distance of f7

is the best (equal to that of M7).

Table 3.8 Distance expansion matrix of h7.

0 448 0 0 0 0 0

0 512 832 0 0 0 0 576 1344 320 0 0 640 1344 256

0 704 640 64 384 64

Table 3.9 Distance expansion matrix of l7.

0 384 0 0 6 22 36

0 516 444 28 128 228 0 582 658 396 604 4 524 776 936 34 436 874 56 392 64

Table 3.10 Distance expansion matrix of r7.

0 384 64 0 0 0 0

0 320 896 128 0 0 0 256 1408 512 64 0 320 1344 576

0 384 960 0 448 64

Chapter 3 DIMs from Binary Vectors 33

Table 3.11 Distance expansion matrix of z7.

0 384 64 0 0 0 0

0 352 832 160 0 0 0 320 1280 576 64 0 352 1280 608

0 384 960 0 448 64

Table 3.12 Distance expansion matrix of f7.

0 384 0 0 0 64 0

0 320 640 0 256 128 0 256 768 640 576 0 192 832 1216

0 192 1152 0 448

64

For n = 8, the distance expansion distribution of f8 is worse than that of r8 and Q8

but is better than that of h8 (see Tables 3.13 ~ 3.16).

Table 3.13 Distance expansion matrix of h8.

0 1024 0 0 0 0 0 0

0 1280 2304 0 0 0 0

0 1600 4160 1408 0 0 0 1920 4992 1920 128

0 2240 3840 1088 128 1792 1664 192 832

128

Chapter 3 DIMs from Binary Vectors 34

Table 3.14 Distance expansion matrix of r8. 0 680 120 112 104 8 0 0

0 576 1704 744 336 216 8 0 568 2856 2552 936 256

0 528 3960 3456 1016 0 744 3920 2504 0 944 2640 0 1024 128

Table 3.15 Distance expansion matrix of Q8.

0 768 256 0 0 0 0 0

0 512 2432 512 128 0 0 0 256 3840 2304 768 0 0 256 4224 3584 896

0 512 3840 2816 0 768 2816 0 1024 128

Table 3.16 Distance expansion matrix of z8 and f8.

0 1024 0 0 0 0 0 0

0 1024 2560 0 0 0 0

0 1024 4096 2048 0 0 0 1024 4608 3072 256

0 1024 4096 2048 0 1024 2560 0 1024 128

For n = 9, we find that large numbers of quantity aggregate on the rightmost column of the distance expansion matrix of f9 (see Table 3.22). Hence, the distance

Chapter 3 DIMs from Binary Vectors 35

expansion distribution of f9 is obviously the best among the six mappings. In addition,

∆1( f9) is the largest among all mappings, including M9. We also notice that r9 and z9

are almost the same except the fourth row (see Table 3.19 and 3.21). The aggregation of quantity in the rightmost column of the distance expansion matrix is a characteristic of fn for n≥ 9 and n is odd (see Tables 3.22, 3.27, and 3.28 for examples). Thus, the distance expansion distribution of fn is better than that of hn, ln, r_n, Q_n, and z_n for n≥ 9 and n is odd. Therefore, we conclude that f_n has better distance expansion distribution than these five previously proposed DPMs or DIMs for n≥ 7 and n is odd. The total distance of fn is also better then that of these mappings, but worse than that of Mn for n≥ 11 and n is odd. However, f_n is a DIM while Mn is not a DIM.

Table 3.17 Distance expansion matrix of h9.

0 2304 0 0 0 0 0 0 0

0 3072 6144 0 0 0 0 0

0 4160 12096 5248 0 0 0

0 5376 16384 9472 1024 0 0 6592 16128 8768 768

256 6272 11520 3456 448 4672 4096 512 1792 256

Table 3.18 Distance expansion matrix of l9.

0 2048 0 0 0 0 6 68 182

0 3076 4092 0 0 40 514 1494 0 4176 8016 2144 126 1646 5396 0 4848 9512 3560 3170 11166

0 4492 7650 5462 14652 4 3200 5496 12804 82 1980 7154

136 2168 256

Chapter 3 DIMs from Binary Vectors 36

Table 3.19 Distance expansion matrix of r9.

0 2048 256 0 0 0 0 0 0

0 1792 6400 1024 0 0 0 0

0 1536 10240 8704 1024 0 0 0 1536 11776 15360 3328 256

0 1536 12544 14848 3328 0 1792 11008 8704 0 2048 7168 0 2304 256 Table 3.20 Distance expansion matrix of Q9.

0 1536 768 0 0 0 0 0 0

0 896 5568 2368 384 0 0 0

0 640 7296 10240 3008 320 0 0 512 8640 15424 7104 576

0 704 9344 16704 5504 0 1024 9792 10688

0 1600 7616 0 2304 256 Table 3.21 Distance expansion matrix of z9.

0 2048 256 0 0 0 0 0 0

0 1792 6400 1024 0 0 0 0

0 1536 10240 8704 1024 0 0 0 1280 12800 13824 4352 0

0 1536 12544 14848 3328 0 1792 11008 8704 0 2048 7168 0 2304 256 Table 3.22 Distance expansion matrix of f9.

0 2048 0 0 0 0 0 0 256

0 1792 5376 0 0 0 0 2048

0 1536 7680 5120 0 0 7168 0 1280 7680 7680 1280 14336

0 1024 6144 6144 18944 0 768 3840 16896 0 512 8704

0 2304 256

Chapter 3 DIMs from Binary Vectors 37

Table 3.23 Distance expansion matrix of r10.

0 4096 1024 0 0 0 0 0 0 0

0 3200 14720 5120 0 0 0 0 0

0 2304 22784 29184 7168 0 0 0

0 2560 24320 56192 21376 3072 0 0 2560 27392 65792 29440 3840

0 2560 30464 55168 19328 0 3328 27904 30208 0 4224 18816 0 5120 512

Table 3.24 Distance expansion matrix of Q10.

0 4096 1024 0 0 0 0 0 0

0 3200 14720 5120 0 0 0 0

0 2304 22784 29184 7168 0 0 0

0 2560 24320 56192 21376 3072 0 0 2560 27392 65792 29440 3840

0 2560 30464 55168 19328 0 3328 27904 30208 0 4224 18816 0 5120 512

Table 3.25 Distance expansion matrix of z10 and f10.

0 5120 0 0 0 0 0 0 0 0

0 5120 17920 0 0 0 0 0 0

0 5120 30720 25600 0 0 0 0

0 5120 38400 51200 12800 0 0 0 5120 40960 61400 20480 1024

0 5120 38400 51200 12800 0 5120 30720 25600 0 5120 17920 0 5120 512

Chapter 3 DIMs from Binary Vectors 38

Table 3.26 Distance expansion matrix of z11.

0 10240 1024 0 0 0 0 0 0 0 0

0 9728 39936 6656 0 0 0 0 0 0

0 9216 67584 78848 13312 0 0 0 0

0 8704 86016 158208 76800 8192 0 0 0 8192 97280 206848 137216 22528 1024

0 8704 97280 214528 133120 19456 0 9216 90112 168960 69632 0 9728 73728 85504 0 10240 46080 0 11264 1024

Table 3.27 Distance expansion matrix of f11.

0 10240 0 0 0 0 0 0 0 0 1024

0 9216 36864 0 0 0 0 0 0 10240

0 8192 57344 57344 0 0 0 0 46080

0 7168 64512 107520 35840 0 0 122880 0 6144 61440 122880 61440 6144 215040 0 5120 51200 102400 51200 263168 0 4096 36864 61440 235520 0 3072 21504 144384 0 2048 54272

0 11264 1024

Table 3.28 Distance expansion matrix of f13.

0 49152 0 0 0 0 0 0 0 0 0 0 4096

0 45056 225280 0 0 0 0 0 0 0 0 49152

0 40960 368640 491520 0 0 0 0 0 0 270336

0 36864 442368 1032192 516096 0 0 0 0 901120 0 32768 458752 1376256 1146880 229376 0 0 2027520

0 28672 430080 1433600 1433600 430080 28672 3244032 0 24576 368640 1228800 1228800 368640 3809280 0 20480 286720 860160 716800 3387392 0 16384 196608 458752 2256896 0 12288 110592 1048576 0 8192 311296

0 53248 4096

Chapter 3 DIMs from Binary Vectors 39

Table 3.29 List of total distance of various DPMs.

n ∆_max hn ln rn Qn zn Mn fn

5 4090 3712 3872 4020* − − 3712 3968

6 20472 18432 − 18432 − 18432 19456* 18432

7 98294 83968 91016 88064 − 88064 94208* 94208*

8 458752 378880 − 413312 409600 393216 458752* 393216

9 2097144 1689600 1911000 1802240 1863680 1802240 1982464 1998848*

10 9437160 7281792 − 8110080 8110080 7863680 9043968* 7863680 11 41943022 31923328 37741432 36330496 − 35127296 40108032* 39321600 12 184549344 138878080 − 154927104 − 150994944 180355072* 150994944 13 805306346 600251520 717371576 677117952 − − 780140544* 746586112

Chapter 4 DPMs from Ternary Vectors 40

Chapter 4

在文檔中 向量至排列之保距映射之建構與分析 (頁 33-50)