Abstract
Let \(\mathbb{C }\) be a code of length \(n\) over an alphabet of \(q\) letters. The descendant code \(\mathsf{desc}(\mathbb C _0)\) of \(\mathbb C _0 = \{\mathbf{c}^1, \mathbf{c}^2, \ldots , \mathbf{c}^t\} \subseteq \mathbb{C }\) is defined to be the set of words \(\mathbf{x} = (x_1, x_2, \ldots ,x_n)\) such that \(x_i \in \{c^1_i, c^2_i, \ldots , c^t_i\}\) for all \(i=1, \ldots , n\). \(\mathbb{C }\) is a \(\overline{t}\)separable code if for any two distinct \(\mathbb{C }_1, \mathbb{C }_2 \subseteq \mathbb{C }\) such that \(\mathbb{C }_1 \le t\), \(\mathbb{C }_2 \le t\), we always have \(\mathsf{desc}(\mathbb{C }_1) \ne \mathsf{desc}(\mathbb{C }_2)\). The study of separable codes is motivated by questions about multimedia fingerprinting for protecting copyrighted multimedia data. Let \(M(\overline{t},n,q)\) be the maximal possible size of such a separable code. In this paper, we provide an improved upper bound for \(M(\overline{2},2,q)\) by a graph theoretical approach, and a new lower bound for \(M(\overline{2},2,q)\) by deleting suitable points and lines from a projective plane, which coincides with the improved upper bound in some places. This corresponds to the bounds of maximum size of bipartite graphs with girth \(6\) and a construction of such maximal bipartite graphs.
Introduction
Let \(n,M\) and \(q\) be positive integers, and \(Q\) an alphabet with \(Q=q\). A set \(\mathbb{C } = \{\mathbf{c}^1,\mathbf{c}^2,\ldots , \mathbf{c}^M\} \subseteq Q^n\) is called an \((n,M,q)\) code and each \(\mathbf{c}^i\) is called a codeword. Without loss of generality, we may assume \(Q=\{0,1,\ldots ,q1\}\).
For any subset of codewords \(\mathbb{C }_0\subseteq \mathbb{C }\), we define the set of \(i\)th coordinates of \(\mathbb{C }_0\) as
and the descendant code of \(\mathbb{C }_0\) as
that is,
Definition 1.1
Suppose \(\mathbb{C }\) is an \((n,M,q)\) code and \(t \ge 2\) is an integer. \(\mathbb{C }\) is a \(\overline{t}\)separable code, or \(\overline{t}\)\(\text{ SC }(n,M,q)\) in short, if for any \(\mathbb{C }_1, \mathbb{C }_2 \subseteq \mathbb{C }\) such that \(\mathbb{C }_1 \le t,\, \mathbb{C }_2 \le t\) and \(\mathbb{C }_1 \ne \mathbb{C }_2\), we always have \(\mathsf{desc}(\mathbb{C }_1) \ne \mathsf{desc}(\mathbb{C }_2)\), that is, there is at least one coordinate \(i\), \(1 \le i \le n\), such that \(\mathbb{C }_1(i) \ne \mathbb{C }_2(i)\).
Let \(M(\overline{t},n,q) = \text{ max } \{M \  \ \text{ there } \text{ exists } \text{ a } \overline{t} \text{SC }(n,M,q)\}\). A \(\overline{t}\)SC\((n,M,q)\) is said to be optimal if \(M = M(\overline{t},n,q)\), and asymptotically optimal if \(\displaystyle \lim _{q \rightarrow \infty } \frac{M}{M(\overline{t},n,q)}=1\).
The study of separable codes is motivated by questions about multimedia fingerprinting which can effectively trace and even identify the sources of pirate copies of copyrighted multimedia data, see, e.g., [6, 19]. It is not difficult to see [6] that identifiable parent property codes [15, 24], frameproof codes [2, 4], perfect hash families [20, 25] and some other structures in digital fingerprinting all imply separable codes.
In multimedia fingerprinting, \(\mathsf{desc}(\mathbb{C }_0)\) consists of all the \(n\)tuples that could be produced by a coalition holding the codewords in \(\mathbb{C }_0\), where the length \(n\) corresponds to the number of orthogonal basis signals in the multimedia content. Since the size \(M\) of \(\overline{t}\)SC\((n,M,q)\) corresponds to the number of fingerprints assigned to authorized users, we should try to construct separable codes with size \(M\) as large as possible, given length \(n\). Cheng and Miao [6] showed that longlength separable codes can be constructed by concatenating shortlength separable codes. This stimulates the investigation of separable codes with length \(n=2\).
In [7] an upper bound on \(M(\overline{2},2,q)\) was derived, and two infinite series of optimal \(\overline{2}\)\((2,M,q)\)SCs were constructed.
Theorem 1.2
[7] For any positive integer \(q\), \(M(\overline{2},2,q) \le qk+t\), where \( k=\lfloor \frac{1+\sqrt{4q3}}{2} \rfloor \), and
Furthermore, \(M(\overline{2},2,q) = qk+t\) if \(q = k^2k+1\) for any prime power \(k1\ge 2\) and \(q = k^2+k\) for any prime power \(k\ge 2\).
In this paper, by using graph theoretical terminologies, we obtain a tighter upper bound on \(M(\overline{2},2,q)\). By using projective geometrical terminologies, we also obtain a lower bound on \(M(\overline{2},2,q)\), parts of which agree with the new derived upper bound. In other words, we construct several new infinite series of optimal \(\overline{2}\)\((2,M,q)\)SCs.
Related combinatorial objects
In order to investigate separable codes, in this section, we describe several related combinatorial structures.
For any \((2,M,q)\) code \(\mathbb{C }\) defined on \(Q=\{0,1,\ldots ,q1\}\), we define \(A_i\) for \(i \in Q\) as \(A_i = \{x_2 \  \ (x_1, x_2) \in \mathbb{C }, \ x_1=i\}\). Obviously, \(A_i \subseteq Q\) holds for any \(i \in Q\), and \(A_0+ A_1 + \cdots + A_{q1}=M\).
Definition 2.1
Let \(K\) be a subset of nonnegative integers, and \(v,b\) be two positive integers. A generalized \((v,b,K,1)\) packing is a pair \((X,\mathbb{B })\) where \(X\) is a set of \(v\) elements and \(\mathbb{B }\) is a set of \(b\) subsets of \(X\) called blocks that satisfy

(1)
\(B \in K\) for any \(B\in \mathbb{B }\);

(2)
every pair of distinct elements of \(X\) occurs in at most one block of \(\mathbb{B }\).
Cheng et al. [7] showed a relationship between separable codes and generalized packings.
Lemma 2.2
[7] There exists a \(\overline{2}\)SC\((2,M,q)\) defined on \(Q\) if and only if there exists a generalized \((q,q,K,1)\) packing \((Q, \{A_0, A_1, \ldots , A_{q1}\})\), with \(K=\{A_0, A_1,\ldots , A_{q1}\}\), and \(M = A_0 + A_1 + \cdots + A_{q1}\).
A generalized \((q,q,\{k\},1)\) packing can be constructed by developing a near difference set. A \((q,k,1)\) near difference set defined on an additively written group \(G\) of order \(G = q\) is a \(k\)subset \(D\) of \(G\) such that the differences \(\{xy \  \ x,y \in D, x \ne y\}\) contains \(k(k1)\) distinct elements of \(G\).
Lemma 2.3
For any integer \(k\ge 2\), let \(q \ge k^2k+1\). If there exists a \((q,k,1)\) near difference set, then there exists a generalized \((q, q, \{k\},1)\) packing.
Proof
Let \(D\) be a \((q,k,1)\) near difference set defined on an additively written group \(G\). For any \(g \in G\), define \(D+g = \{x+g \  \ x \in D\}\) and \(\mathbb B = \{D+g \  \ g \in G\}\). Then \((G, \mathbb B )\) is the desired generalized \((q, q, \{k\},1)\) packing.\(\square \)
Near difference sets are not easy to construct. However, a \((k^2+k+1,k,1)\) near difference set always exists [23] for any prime power \(k\). This Singer difference set generates a generalized \((k^2+k+1, k^2+k+1, \{k\},1)\) packing, which corresponds to an optimal \(\overline{2}\text{ SC }(2,(k+1)(k^2+k+1),k^2+k+1)\) described in Theorem 1.2.
Given a generalized \((v,b,K,1)\) packing \((Q,\mathbb B )\), we can define its associated elementblock graph as the bipartite graph \(G_{Q,\mathbb B }\) with vertex partition \(Q\) and \(\mathbb B \) such that \(x\in Q\) is adjacent to \(B \in \mathbb B \) if and only if \(x\in B\). It is clear that the corresponding elementblock graph of a generalized \((v,b,K,1)\) packing \((Q,\mathbb B )\) is a \(C_4\)free bipartite subgraph of \(K_{v,b}\), because any pair of distinct elements of \(Q\) can occur in at most one block of \(\mathbb B \). In other words, the girth of this bipartite graph is at least \(6\), where the girth of a graph is the length of a shortest cycle contained in the graph.
Zarankiewicz numbers [27] involve bounds on the maximum number of edges in a bipartite graph without a particular subgraph. We denote by \(z(m,n;s,t),\, m\le n\) and \(s\le t\), the maximum number of edges in a subgraph of \(K_{m,n}\) that does not contain a subgraph isomorphic to \(K_{s,t}\). In particular, when \(m=n\) and \(s=t\), simply put \(z(n;t)=z(n,n;t,t)\). It is clear that \(z(q;2)\), which is the maximum size of a \(C_4\)free bipartite subgraph of \(K_{q,q}\), is equals to \(M(\overline{2},2,q)\) by Lemma 2.2. Meanwhile, GarcíaVázquez et al. [11] stated that any \(C_4\)free bipartite subgraph of \(K_{q,q}\) with size \(z(q;2)\) must have girth \(6\). Therefore, our problem is equivalent to finding the maximum size of bipartite graphs with girth \(6\), where the size of a graph refers to the number of edges it contains, and constructing such maximal bipartite graphs.
We can see our problem in one more way. Given a generalized \((q,q,K,1)\) packing \((Q,\mathbb B )\), if we define two elements of \(Q\) are adjacent in \(B \in \mathbb B \) if they occur in the same block \(B\), then each block can be seen as a clique of order \(B\) belonging to \(K\). Since each pair of distinct elements of \(Q\) occurs in a block of \(\mathbb B \) at most once, this generalized \((q,q,K,1)\) packing can be viewed as a packing of \(K_q\) by \(q\) cliques of orders belonging to \(K\). Therefore, in order to evaluate \(z(q;2)=M(\overline{2},2,q)\), it is sufficient to pack \(K_q\) by \(q\) cliques so that the sum of order of the \(q\) cliques is maximum.
It is well known [3] that \(z(q;2) \le (q+q \sqrt{4q3})/2\) and the equality holds when \(q=k^2+k+1\) for any prime power \(k\). Goddard et al. [12] found the exact values of \(z(q;2)\) for \(q \le 10\). Very recently, Damásdi et al. [8] also found the exact values of \(z(q;2)\) for \(q = k^2+k2, k^2+k1\) with \(k\) being a prime power, among others. Theorem 1.2 is an improvement of the results made by Cheng et al. [7]. It is also known [3] that if \(q\) is sufficiently large then
In particular, \(\displaystyle \lim _{q \rightarrow \infty } \frac{z(q;2)}{q^{3/2}}=1\). For the uptodate information on Zarankiewicz numbers, the reader is referred to [8].
Upper bound
Bipartite graphs with high girth and their related graphs have been extensively investigated, see, e.g., [1, 8–11, 13, 16–18, 21, 22, 26]. We start this section with the following proposition.
Proposition 3.1
[5] Suppose \((X,\mathbb B )\) is a generalized \((v,b,\{k,k+1\},1)\) packing, for some \(k\), with \(\mathbb B = \{B_1, B_2, \ldots , B_b\}\). If \(\displaystyle {\left( \begin{array}{l}v\\ 2\end{array}\right) }  \sum _{i=1}^b {\left( \begin{array}{c}B_i\\ 2\end{array}\right) } < k\), then \(G_{X,\mathbb B }\), the elementblock graph of \((X,\mathbb B )\), is a \(C_4\)free subgraph of \(K_{v,b}\) with maximum size.
If \(K_q\) can be packed by \(q\) cliques \(K_{x_1}, K_{x_2}, \ldots , K_{x_q}\) with leave \(L\), where \(x_i\le x_j\) for \(1 \le i < j \le q\), then we say \(K_q\) admits a feasible \((x_1,x_2,\ldots , x_q)\) packing with leave \(L\). For convenience, we replace \((x_1,x_2,\ldots , x_q)\) packing by \((k^{qt},(k+1)^t)\) packing when
for some \(k\in \mathbb N \) and \(1\le t\le q\). For any \((k^{qt},(k+1)^t)\) packing \(\mathcal P \) of \(K_q\), we have
which implies \(k\le \frac{1+\sqrt{4q3}}{2}\). In order to maximize \(\sum _{i=1}^q x_i\) which subjects to an \((x_1,x_2,\ldots , x_q)\) packing, Proposition 3.1 promises to consider a feasible \((k^{qt},(k+1)^t)\) packing with \(k = \lfloor \frac{1+\sqrt{4q3}}{2} \rfloor \) and \(L<k\). Therefore, our objective is to find the maximum index \(t\). Note that \(k = \lfloor \frac{1+\sqrt{4q3}}{2} \rfloor \) implies \(k^2k+1\le q< k^2+k+1\). In this section, we investigate \(z(q;2)\) by fixing the index \(k\) and then classifying \(q\), from \(k^2k+1\) to \(k^2+k\), into several cases. The following Theorem 3.2 is contained in Theorem 1.2.
Theorem 3.2
[3, 7] For any prime power \(k1 \ge 2,\, z(k^2k+1;2)=k^3k^2+k\). For any prime power \(k \ge 2,\, z(k^2+k;2) = k^3+2k^2\).
Theorem 3.3
For any \(k^2+1\le q\le k^2+k2\) and \(k\ge 2\), we have
Proof
Let \(q=k^2+ks,\, s=2,3,\ldots ,k1\). Assume \(\mathcal P \) is a \((k^{qt},(k+1)^t)\) packing of \(K_q\), where \(0\le t\le q1\). We claim by contradiction that \(t\le \lfloor \frac{(k1)q}{(k+1)^2(q+1)} \rfloor \). That is, suppose \(t > \lfloor \frac{(k1)q}{(k+1)^2(q+1)} \rfloor \).
For \(i\ge 0\), let \(r_i\) be the number of vertices that is contained in exactly \(i\) cliques of order \(k+1\) in \(\mathcal P \). Since the degree of each vertex is \(k^2+ks1\), trivially \(r_i=0\) for all \(i> k\). We now claim \(r_k=0\). Suppose not, that is, there exists a vertex \(v\) contained in exactly \(k\) cliques of order \(k+1\), say \(C_1, C_2, \ldots , C_k\). Let \(A=\displaystyle \{v\} \cup \bigcup _{i=1}^k V(C_i)\) and \(B=V(K_q)\setminus A\), where \(V(G)\) is the vertex set of graph \(G\). Since there is no other subgraph isomorphic to \(K_{k+1}\) out of \(A\) except \(C_1, C_2, \ldots , C_k\), each of the remaining cliques of order \(k+1\) must contain at least one vertex in \(B\). That is, each of such cliques needs at least \(k\) edges between \(A\) and \(B\). Therefore, we have at most \(k+ \lfloor \frac{(k^2+1)(ks1)}{k} \rfloor \) cliques of order \(k+1\). Thus,
This implies \(ks^2ksk+s<0\), so \(ks(s1) \le ks < k\), that is, \(s(s1) <1\), which contradicts \(2\le s\le k1\). So \(r_k=0\).
Consider the number of ordered pairs \((v,C)\), where \(v\) is a vertex in the clique \(C\) of order \(k+1\) in \(\mathcal P \). Under our assumption, there are exactly \(t\) cliques of order \(k+1\), then
This implies that
so
Now, we drop all the \(t\) cliques of order \(k+1\) from \(K_q\). Denote by \(G\) the remaining subgraph. We again consider the number of ordered pairs \((v^{\prime },C^{\prime })\), where \(v^{\prime }\) is a vertex in the clique \(C^{\prime }\) of order \(k\) in \(\mathcal P \). On one hand there are exactly \(qt\) cliques of order \(k\), and on the other hand there are exactly \(r_{i}\) vertices of degree \(q1ki\), for \(i=0,1, \ldots , k1\). Since the vertex of degree \(q1ki\) can be contained in at most \(\frac{q1ki}{k1}\) cliques of order \(k\), we have
and thus
Finally, (2) and (4) imply that \(t\le \frac{q(k1)}{k+s}=\frac{(k1)q}{(k+1)^2(q+1)}\), a contradiction to the hypothesis. Thus we complete the proof.\(\square \)
Theorem 3.4
For any \(q = k^2\) with \(k\ge 2\), we have
Proof
Assume \(\mathcal P \) is a \((k^{qt},(k+1)^t)\) packing of \(K_q\). For \(i\ge 0\), let \(r_i\) be the number of vertices that is contained in exactly \(i\) cliques of order \(k+1\) in \(\mathcal P \). Since \(q=k^2\), we have \(r_i=0\) for all \(i\ge k\). Similar to the Proof of Theorem , we first consider the number of ordered pairs \((v,C)\), where \(v\) is a vertex in the clique \(C\) of order \(k+1\) in \(\mathcal P \). Then after dropping those cliques of order \(k+1\), we consider the number of ordered pairs \((v^{\prime },C^{\prime })\), where \(v^{\prime }\) is a vertex in the clique \(C^{\prime }\) of order \(k\) in \(\mathcal P \). Note that in the remaining graph after dropping \(t\) cliques of order \(k+1\), there are exactly \(r_i\) vertices of degree \(k^2ik1\). Then we have
which implies \(t\le r_0\). This concludes that the \(t\) cliques of order \(k+1\) are out of at most \(k^2t\) vertices somewhere in \(\mathcal P \). We immediately have
That is,
Since \(t \le k^2\), the above inequality is true only when \(t\le \frac{(3k^2+k1)\sqrt{5k^4+6k^3k^22k+1}}{2}\). Hence we complete the proof.\(\square \)
Theorem 3.5
For any \(k^2k+2\le q\le k^21\) and \(k\ge 2\), we have \(z(q;2)\le qk\).
Proof
Let \(q=k^2s\), where \(s=1,2,\ldots ,k2\). Assume \(\mathcal P \) is a \((k^{qt},(k+1)^t)\) packing of \(K_q\). Suppose \(t\ge 1\). Define \(G\) to be the graph by dropping one of the cliques of order \(k+1\), say \(\widehat{K}\), from \(K_q\). Let \(A\subseteq V(G)\) be the collection of vertices whose degree is equal to \(q1k\), and \(B=V(G)\setminus A\). Note that \(A=k+1\) and \(B=qk1\). Now, consider the number of ordered pairs \((v,C)\), where \(v\) is a vertex in the clique \(C\) in \(\mathcal P \) different from \(\widehat{K}\). Notice that for each \(v\in A,\, \mathsf{deg}_G(v)=k^2s1k=(k1)^2+(ks2)\), then \(v\) is contained in at most \(k1\) cliques different from \(\widehat{K}\). Similarly, each vertex in \(B\) can be contained in at most \(k\) cliques. By counting the number of pairs \((v,C)\), we have
This implies that \(t \le 0\), a contradiction occurs. Thus the result follows.\(\square \)
Lower bound
Now we derive a lower bound on \(z(q;2)= M(\overline{2},2,q)\) via projective planes. A projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties:

(1)
Given any two distinct points, there is exactly one line incident with both of them.

(2)
Given any two distinct lines, there is exactly one point incident with both of them.

(3)
There are four points such that no line is incident with more than two of them.
Clearly, a projective plane of order \(k\) is a generalized \((k^2+k+1,k^2+k+1,\{k+1\},1)\) packing \((X, \mathbb{B })\) in which every pair of distinct elements of \(X\) occurs in exactly one block of \(\mathbb{B }\). It is wellknown [14] that a projective plane of order \(k\) always exists for any prime power \(k\).
Theorem 4.1
For any prime power \(k \ge 2\), let \(k^21 \le q \le k^2+k1\). Then there exists a generalized \((q,q,\{k,k+1\},1)\) packing, \((X^{\prime },\mathbb B^{\prime } )\), with \(X^{\prime } = \mathbb B^{\prime }  = q\) such that exactly \(k^3k^2kqk+2q+1\) blocks out of \(\mathbb B^{\prime } \) are of size \(k\). That is,
Proof
We start from a projective plane of order \(k\), \((X,\mathbb B )\). Note that \(X=\mathbb B =k^2+k+1\), and for any \(B \in \mathbb B ,\, B=k+1\). Pick an arbitrary point \(a \in X\) and an arbitrary line \(B^* = \{ x_1,x_2,\ldots , x_{k+1} \}\in \mathbb B \) which does not contain the point \(a\). For each \(i=1,\ldots , k+1\), let \(B_i \in \mathbb{B }\) be the line containing the points \(a\) and \(x_i\). Let \(2 \le s \le k+2\). Dropping \(s\) lines \(B^*, B_1, \ldots , B_{s1}\) and \(s\) points \(a, x_1,\ldots , x_{s1}\) from \((X,\mathbb B )\), we obtain a generalized \((q,q,\{k,k+1\},1)\) packing, \((X^{\prime },\mathbb B^{\prime } )\), with \(q=k^2+k+1s,\, X^{\prime }= X \setminus \{a,x_1,\ldots ,x_{s1}\},\, \mathbb B^{\prime } = \mathbb{B } \setminus \{ B^*, B_1, \ldots , B_{s1}\}\), having \({\Delta } = (s1)(k1)+(k+1s+1)=k^3k^2kqk+2q+1\) blocks of size \(k\) and \(k^2+k+1{\Delta }s\) blocks of size \(k+1\). Therefore, \(z(q;2) \ge k{\Delta }+(k+1)(k^2+k+1\Delta s) = 2qkk^3+k^2+kq1\).\(\square \)
Applying Theorems 1.2, and 3.5, we immediately have the following result.
Corollary 4.2
For any prime power \(k \ge 2,\, z(k^21;2)= k^3k,\, z(k^2+k2;2)= k^3+2k^24k+1,\, z(k^2+k1;2)= k^3+2k^22k\).
We remark that Damaásdi et al. [8] obtained independently the same results for \(q=k^2+k2, k^2+k1\) in Corollary 4.2. It is easy to verify that the corresponding \(\overline{2}\)\(\text{ SC }(2,M,q)\)s constructed in Theorem are asymptotically optimal for all \(k^21 \le q \le k^2+k1\) with prime power \(k\). The lower bound described in Theorem is better than \(q^{3/2}  q^{4/3}\) described in [3] for any prime power \(k\).
Summary
The main results in the previous sections can be summarized in the following theorem.
Theorem 5.1
For any positive integer \(q,\, M(\overline{2},2,q) \le qk+t\), where \(k=\lfloor \frac{1+\sqrt{4q3}}{2} \rfloor \), and
Furthermore, \(M(\overline{2},2,q) = qk+t\) if \(q=k^2k+1\) for any prime power \(k1 \ge 2\), and \(q=k^21, k^2+k2,k^2+k1, k^2+k\) for any prime power \(k \ge 2\).
The above Figs. 1 and 2 illustrate our improvement on the upper bound of \(M(\overline{2},2,q)\). Figure 1 depicts the known upper bound given in [7] and the new upper bound given in Theorem 5.1 when \(k=12\), while Fig. 2 depicts those upper bounds when \(q=k^2\). It can be seen that our new upper bound is much tighter than the known upper bound.
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Acknowledgments
The interesting paper [8] was drawn to the authors’ attention by one of the two anonymous reviewers. The authors express their sincere thanks to the reviewers for their valuable comments and suggestions in revising this paper. Cheng is supported by Guangxi Natural Science Foundation under Grant No. 2013GXNSFCA019001, by the general scientific research project of the Education Department of Guangxi Province (Grant No. 2013YB039 ), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Fu and Lo are supported by NSC 1002115M009005MY3. Miao is supported by JSPS GrantinAid for Scientific Research (C) under Grant No. 24540111.
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Communicated by C. J. Colbourn.
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Cheng, M., Fu, HL., Jiang, J. et al. New bounds on \(\bar{2}\)separable codes of length 2. Des. Codes Cryptogr. 74, 31–40 (2015). https://doi.org/10.1007/s1062301398499
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Keywords
 Multimedia fingerprinting
 Separable code
 4Cycle free bipartite graph
 Zarankiewicz number
 Projective plane
Mathematics Subject Classification
 94B25
 94A62
 05B40
 05C51
 05B25