New Bounds on 2-Frameproof Codes of Length 4

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. (ese digital fingerprints are generally denoted as codewords in Q, where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. (us, no pair of users can frame a user who is not a member of the coalition. (is paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that |C|≤ 2q2 − 2q + 1 when q is odd and q> 10.


Introduction
In order to protect a digital content, a distributor marks each copy with a codeword. is marking discourages users from releasing an unauthorized copy, since a mark allows the distributor to detect any unauthorized copy and trace it back to the user. However, a coalition of users may detect some of the marks, namely, the ones where their copies differ. us, they can forge a new copy by changing these marks arbitrarily. To prevent a coalition of users from "framing" a user outside the coalition, Boneh and Shaw [1] defined the concept of frameproof codes. A w-frameproof code has the property that no coalition of at most w users can frame a user not in the coalition. Frameproof codes are defined as follows.
Let q and n be positive integers. Let Q be a set of size q, and let C ⊆ Q n be a set of words of length n over the alphabet Q. Each codeword x ∈ C can be represented as x � (x 1 , x 2 , . . . , x n ), where x i ∈ Q for all i ∈ 1, 2, . . . , n { }. e set of descendants of X ⊆ C, desc(X), is defined as desc(X) � c ∈ Q n : c i � x i for some x ∈ X .
(1) Definition 1. Let w be an integer such that w ≥ 2. A w-frameproof code is a subset C ⊆ Q n such that for all X ⊆ C with |X| ≤ w, we have that desc(X) ∩ C � X.
As the number of codewords in the code corresponds to the number of authorized users in the digital copyright protection system, one of the classical question regarding frameproof codes is what is the largest cardinality M n,w (q) of a w-frameproof code of length n over an alphabet of size q? is paper concentrates on the special case when w � 2, n � 4, and q is large.
Theorem 1 (see [2]). Let q, n, and w be positive integers such that q ≥ 2 and 2 ≤ n ≤ w. en, Theorem 2 (see [2]). Let n be a positive even integer such that n ≥ 4. Let m be a prime power such that m ≥ n + 1. Let q � m 2 + 1. ere exists a q-ary 2-frameproof code of length n of size Theorem 3 (see [4]).
ere exists a q-ary w-frameproof code of length w + 2 of size for all odd q, when w � 2, and for all q ≡ 4(mod 6), when w � 3.
For 2-frameproof codes of length 4, eorem 3 only gives lower bound for odd q. For even q, we first use only q − 1 symbols to construct a (q − 1)-ary 2-frameproof code of size 2(q − 2) 2 + 1. en, add a codeword (α, α, α, α) to the code, where α is the unused symbol. us, the following result on the lower bound of M 4,2 (q) is obtained.
Corollary 1 (see [4]). For any positive integer q, In 2003, Blackburn proved the following upper bound of w-frameproof codes.
Theorem 4 (see [2]). For any positive integers q, n, and w ≥ 2, if C is a q-ary w-frameproof code of length n, then where r is a unique integer in 0, 1, 2, . .
When n � 4 and w � 2, the following result on the upper bound of M 4,2 (q) is obtained.

Corollary 2. For any positive integer q,
Much later on, in 2019, Cheng et al. proved the following theorem, which is the best previously known result on the upper bound of 2-frameproof code of length 4.
Theorem 5 (see [5]). For any positive integer q > 48, if C is a q-ary 2-frameproof code of length 4, then Our result is the improved version of eorem 5. We aim to prove the following theorem. Theorem 6. For any odd positive integer q > 10, if C is a qary 2-frameproof code of length 4, then We analyze the combinatorial structure of a code, setting up an optimization problem, deriving some constraints, and solving this optimization problem to obtain eorem 6. e gap between the lower bound of odd and even q in Corollary 1 is the key motivation for proving the main result. e rest of this paper is ordered as follows. In Section 2, the essential·notations are defined. Necessary conditions of a q-ary 2-frameproof code of length 4 are also stated. In Section 3, the proof of eorem 6 is provided. We conclude the result in the last section.

Preliminaries
In this section, we define some notations and state relating lemmas that are useful for proving the main theorem. Let , and for any non-empty set I ⊆ [4], For Remark. It is easy to see that for any nonempty subsets I, J ⊂ [4], the following conditions hold: Lemma 1. Let C be a q-ary 2-frameproof code of length 4. For any x � (x 1 , x 2 , x 3 , x 4 ) ∈ C and any nonempty subset Hence, x ∈ desc( y, z ). is contradicts the 2-frameproof property of C. For convenience, we set up some parameters. Let From Lemma 1, we can see that Hence, we can deduce upper bound of C from the upper bounds of |U 1,2,3 From the definition of as required.
We use this lemma to find constrains on the upper bound of |V i,j,k { } |. en, after Section 2.3, we eliminate the term qh 4 before proving the main theorem. is section gives the upper bound of the second component of equation (11). It gives the same results as [5]. We put it here for completeness. We find an upper bound of |V 3,4 { } | by counting elements in f 1,2,3 { } (C).

Lemma 3. Suppose C is a q-ary 2-frameproof code of length 4. en,
forbidden values in f 1,2,3 forbidden values in f 1,2,3 { } (C). However, the triple in the form of (f 1 and c ∈ Q will be counted twice. us, together, V 2,3,4 values from f 1,2,3 (17) and (20). us, we obtain at least different forbidden values in f 1,2,3 { } (C) from this step. Hence, From equations (22) and (23), we have (24) us, We use this lemma to find constrains on the upper bound of |V i,j,k { } | in the next section.
is section aim to eliminate the third component of equation (11). Recall that we define h 1 � |V 2,3,4 Proof. From equation (11), Lemma 2 and Lemma 3, we have From Corollary 1, we have that there always exists a qary 2-FP code of size 2(q − 1) 2 + 1 for any positive odd integer q. So, we have which can be rewritten as Note that if h 1 � q or h 2 � q, there are at most q codewords in C. We then only consider the case 0 ≤ h 1 and h 2 ≤ q − 1. We have Substitute in equation (28), we obtain us, when q > 10.
Since it is impossible to have a single codeword in C, that is, nonunique under i, j, k , then h ℓ ≠ 1 for all ℓ ∈ [4]. erefore, if h 1 ≠ 0, then h 1 ≥ 2. us, with the maximal of h 1 , we have h 2 � h 3 � h 4 � 0. Furthermore, there are only 3 possible cases for h 1 , which are h 1 � 3, h 1 � 2, and h 1 � 0. □ Remark. Note that the condition q > 10 can be removed by substituting corresponding values into equations (28) and (32) repeatedly until a contradiction is reached.

Main Results
In this section, we aim to prove eorem 6, which is the main theorem.
By applying the condition h 2 � h 3 � h 4 � 0 from Lemma 4 to equations (11) and (26), we obtain new equations: We also obtain the following corollary from Lemma 2. Corollary 3. Suppose C is a q-ary 2-frameproof code of length 4. en, Using equations (33) and (34) and Corollary 3, we now prove eorem 6. Proof So, by equations (33) and (36) and Corollary 3, we have us, by Corollary 1, Hence, 4 International Journal of Mathematics and Mathematical Sciences (39) us, we only have yet to consider the case that |f 3,4 Since |C| must be an integer, |C| ≤ 2q 2 − 2q + 1 as required.
Example 1 gives a 3-ary 2-frameproof of size 12 � 2q 2 − 2q for q � 3, which is very close to the obtained upper bound. However, things could be different for larger q.

Conclusion
In this paper, we investigate the bounds of 2-frameproof codes with length 4 by observing the structure of a code. e improvement of the upper bound for the case of odd q is derived from the difference between the known lower bound of odd and even q. e paper shows that |C| ≤ 2q 2 − 2q + 1 in the case when q is odd and q > 10. en, if q is large, 2q 2 − 4q + 3 ≤ M 4,2 (q) ≤ 2q 2 − 2q + 1 when q is odd and 2q 2 − 8q + 10 ≤ M 4,2 (q) ≤ 2q 2 − 2q + 7 when q is even. Example 1, for q � 3, suggests that the upper bound might be tighter than the lower bound; however, the case of larger q is yet to be determined.

Data Availability
No data were used to support this study since all proofs are included in the manuscript.

Conflicts of Interest
e author declares that there are no conflicts of interest.