Constructions for hash families have been extensively investigated by many re-searches. Here, we assume the existence of certain hash families and use them to construct secure frameproof codes. We first construct small codes and use them as the initial seed to construct bigger ones.
We will use sandwich free families, perfect hash families, and separating hash families to construct SFP codes. Note that in the construction the unreadable marks are unnecessary for discussion. Before doing so, we present a direct con-struction and a recursive concon-struction of SFP codes which explains the idea of recursive construction.
Theorem 5.14. For any integerw > 2, there is a w− SF P 2w−1w−1 , 2w
.
Proof. Recall the representation of incidence matrix defined in Section 3.5of set systems. Let the codeC be built from an incidence matrix whose first row contains all 1s and the remaining columns corresponds to B which is the set of subsets B1, . . . , BN, where Bi contains all possible (w − 1) choices out of (2w − 1) elements, yielding N = 2w−1w−1
. We will show that C =
Example 5.5. Using the above method, a 3-SFP(10,6) code can be constructed and interpreted as an incidence matrix as follows:
M(C) =
A recursive construction can be provided in a similar way.
Theorem 5.15. For any integerw > 2, there is a w− SF P 2 2w−1w−1
, 2w + 1 .
CHAPTER 5. CONSTRUCTIONS OF SFP CODES 41 Proof. LetC be the code defined in Theorem5.14. Denote byM(C) the incidence matrix ofC of dimension 2w× 2w−1w−1
Then, it is not hard to say that M can serve as the incidence matrix of a w − SF P 2 2w−1w−1
, 2w + 1 .
Next, we formulate the SFP codes in terms of hash families. Recall the defini-tions of set systems and sandwich free families defined earlier in Section3.5and Section3.6: codewords inC have jth bit equal to 0. The conclusion follows.
Based on the lemma, we restate Theorem3.5and prove it now.
Theorem 5.16. A w− SF P (N, n) exists if and only if there exists a (w, w) − SF F (N, n).
Proof. Suppose that(X,B) is a set system. It suffices to say that (X, B) is not a (w, w)− SF F if and only if there is a set W ⊆ X such that associated(n, N)-code, the two conditions above are equivalent to
descw(C1)∩ descw(C2)6= ∅.
by Lemma5.1.
The following two theorems are given earlier in Chapter 3 which will be used now.
Theorem 5.17. A(N, n, q)-code, C, is a w−SF P code if H(C) is an P HF (N; n, q, 2w), wheren > 2w.
Theorem 5.18. A(N, n, q)-code, C, is a w− SF P code if and only if H(C) is anSHF (N; n, q, w, w), where n > 2w.
These theorems tell us that if we can find the constructions for PHF or SHF, we have equivalently the SFP codes as well. We now examine the recursive con-struction.
CHAPTER 5. CONSTRUCTIONS OF SFP CODES 43 In [4], a recursive construction of perfect hash families is discussed in order to provide an infinite class of prefect hash families. They are stated in the following two theorems.
Proof. Recall the definition of difference matrices mentioned in Section3.4. De-note D = (di,j) by the rule di,j = ij mod n0, 0 6 i 6 w2
-difference matrix which can be embedded into the originalP HF (N0; n0, m, w) to yield a bigger
P HF w2 + 1
N0; n20, m, w .
One nice thing about Theorem5.20is that it can be iterated.
Theorem 5.21. Suppose there exists a P HF (N0; n0, m, w), and suppose that
In order to iterate, we need two seeds as the following:
Example 5.6. There exists aP HF (7; 7, 4, 4) as follows: inci-dence matrix can be depicted as follows:
Then, with Theorem5.19 and Theorem5.21 in mind, we have the following infinite classes of 2-SFP codes.
Theorem 5.22. There exists a2− SF P (3 · 7j+1, 72j) for all j > 0.
Proof. We have Example5.6as a initial seed, and we iterate by Theorem 5.21to get an infinite class of P HF
4 2
+ 1j
7; 72j, 4, 4
= P HF
7j+1; 72j, 4, 4 for all j > 0. On the other hand, since a 2− SF P (3, 4) exists by Example5.7, we have an infinite class of2− SF P (3 · 7j+1, 72j) by Theorem5.19.
If we use separating hash families instead of perfect hash families, we also have a similar recursive construction.
Theorem 5.23. If there exists an(w1, w2)−SF F (v, m) and an SHF (N; n, m, w1, w2), then there exists an(w1, w2)− SF F (vN, n).
The proof is similar as before. Also in [4], a similar recursive construction for providing infinite class of separating hash families is provided as follows:
Theorem 5.24. Suppose there exists anSHF (N0; n0, m, w1, w2), where gcd(n0, (w1w2)!) = 1. Then there exists an SHF
(w1w2+ 1)jN0; n20j, m, w1, w2 for any integerj > 0.
An initial seed of separating hash families can be provided below:
Example 5.8. There exists anSHF (3; 7, 4, 2, 2) as follows:
1 1 2 2 3 3 4 2 1 1 2 4 3 3 1 2 1 2 3 4 3
Combining the seeds served by Example5.7and Example5.8leads to:
Theorem 5.25. There exists a2− SF P (9 · 5j, 72j) for all j > 0.
Proof. From Theorem5.24and Example5.8, we have an infinite class ofSHF (3· 5j; 72j, 4, 2, 2) for all j > 0. Since a 2− SF P (3, 4) exists by Example 5.7, the conclusion follows by Theorem5.23.
CHAPTER 5. CONSTRUCTIONS OF SFP CODES 45 Here the code rate is
R = N−1log n
= 1
9· 5j log 72j
= 2jlog 7 9· 5j
= log 7 9
2 5
j
which still tends to zero asj goes to infinity.
Also, the asymptotic behavior of code length isN =O
(log7n)log27 . A more general result forw > 2 can be provided in a similar fashion.
Theorem 5.26 (Stinson [31]). Letw > 2. Then there exists an w− SF P
2 2d−1d−1
· (w2+ 1)j, (2d + 1)2j
for allj > 0 and d > w such that gcd (2d + 1, (w2)!) = 1.
The proof is similar to the one of Theorem 5.20 combining the existence of w− SF P 2 2w−1w−1
, 2w + 1
in Theorem5.15.
The following result is an immediate corollary of Theorem5.26.
Corollary 5.2. For anyw > 2, there exists an explicit construction for an infinite class ofw− SF P (N, n) where N = O
(log n)log2(w2+1)
It is important that we choose our seeds as best as possible. Moreover, in [5, 32], more new constructions for perfect hash families and separating hash families are established using orthogonal arrays and Latin rectangles as follows.
Theorem 5.27. For any positive integersm and w such that w 6 m, there exists an infinite class ofP HF (N; n, m, w) for which N isO
(w2)log∗n(log n) .
Note that log∗ : N 7→ N is a function growing very slowly and is defined recursively as
log∗1 = 1
log∗n = log∗(⌈log n⌉) + 1, if n > 1.
Example 5.9. log∗1010 = log∗10 + 1 = log∗1 + 1 + 1 = 1 + 1 + 1 = 3.
Theorem 5.28. For any positive integers m, w1 andw2, there exists an infinite class ofSHF (N; n, m, w1, w2) for which N isO
(w1w2)log∗n(log n) .
This gives immediately the following consequence.
Corollary 5.3. For any positive integers m, w, there exists an infinite class of w− SF P (N, n, q) for which N is O
(w2)log∗n(log n) .
Here the code rate is
R = N−1log n∼ log n
(w2)log∗n(log n) = 1 (w2)log∗n which would tend to zero however much slower than the previous one.
The proofs of Theorem5.27and5.28are beyond the scope of our thesis and can be found in [32].
CHAPTER 5. CONSTRUCTIONS OF SFP CODES 47