Converted from q-Ary Matrices

Du et al. [19] gave a construction of the (H_r : d)-disjunct matrices by first constructing a q-ary matrix Q and then converting it to a binary matrix M. Let f_j(e) denote the set of q-ary entries in row j collected from the columns associated with the edge e ∈ E. Then Q has the property that for any d + 1 edges e₀, e₁, · · · , e_d, conversion is to replace row j in Q by c_j rows, each of which labeled by the set {(j, f )} where f is a distinct element in the set {f_j(e) : e ∈ E}. For row {(j, f )} in the converted matrix M, every column in e with f_j(e) = f (there can be more than one such edge e) has a 1-entry and all other columns have a 0-entry. They proved that such a matrix M converted from a q-ary matrix Q is (H_r: d)-disjunct. They also gave a construction of Q = [qij] with drm + 1 rows and q^m+1 columns each representing a degree-m polynomial pv(x) in GF (q), where v ∈ V and q is a prime power ≥ drm + 1, of tests in the converted matrix M is

drm+1 exists at least a row x_j in M, converted from the row x in Q, in which all columns in e0 have 1-entries while each Ci has 0-entries, 1 ≤ i ≤ d. Hence ∩e0 6⊆ d)-disjunct matrix with n columns. Similarly, we define t(n, d, r] as the minimum number of rows required for a (d, r]-disjunct matrix with n columns. Existing results on t(q, d, r] (see [50] for an example) show that it is less than

µq + r − 1 r

¶

in general or at least asymptotically. Thus, we have

Theorem 5.2.1. t(q^m+1, d : Hr) ≤ (drm + 1) · t(q, d, r].

When H is the complete r-graph, M is (K_r : d)-disjunct. By Theorem 5.1.2, M is also (d, r]-disjunct. Then we have

Corollary 5.2.2. t(q^m+1, d, r] ≤ (drm + 1) · t(q, d, r].

Corollary 5.2.2 is the same result [20] as given by D’yachkov, Vilenkin, Macula and Torney on the construction of (d, r]-disjunct matrices using the MDS-code. The incidence matrix of the MDS-code with parameters (q, k, t) is a q-ary matrix of size t × q^k and the Hamming distance of any pair of columns is d = t − k + 1. Lemma 5.2.3 arises from the definition of the MDS-code.

Lemma 5.2.3. (Sagalovich [46]). If q^k ≥ d + r and t ≥ dr(k − 1) + 1, then any MDS-code with parameters (q, k, t) has the property that for any d + r columns C₁, C₂, · · · , C_d+r, there exists a row where the set of entries over C₁, C₂, · · · , C_r and the set of entries over C_r+1, · · · , C_d+r are disjoint.

D’yachkov et al. used the Reed-Solomon q-ary code, which is also an MDS-code, to get a (drm + 1) × q^m+1 q-ary matrix with the property that described in Lemma 5.2.3. Then they also use a t⁰× q (d, r]-disjunct matrix to transform the q-ary matrix to binary one. The requirement of this q-ary matrix is seemingly different from that given by Du et al., though the latter also corresponds to an MDS-code. Chen, Du and Hwang [10] proved that the requirements of the two q-ary matrices are equivalent.

Let e₀, e₁, · · · , e_d be any d + 1 complexes. Set {C₁, C₂, · · · , C_r} = e₀ and C_r+i ∈

Stinson and Wei [49] first gave an error-tolerant version of the (d, r]-disjunct matri-ces. Recall that a matrix is (d, r; z]-disjunct if for any d + r columns C₁, C₂, · · · , C_d+r,

i.e., there exist at least z rows in which each of the r designated columns has a 1-entry and each of the other d columns has a 0-entry. For a (d, r; 2e + 1]-disjunct matrix, if the number of errors is less than e, then we can identify the up-to-d positive complexes consisting of at most r items. It is because that each negative complex appears in at least (2e + 1) − e = e + 1 negative pools, while each positive complex in at most e negative pools (due to errors). Therefore we can separate the negative complexes from the positive ones.

Still, using a (d, r; 2e + 1]-disjunct matrix, we can identify all positive complexes as well even if the composition of given complexes is not specified. The decoding algorithm is similar to GENERAL COMPLEX ALGORITHM except replacing t^V₀(X) = 0 by t^V₀(X) ≤ e.

(d, r, e)-GENERAL COMPLEX ALGORITHM

0 use a (d, r; 2e + 1]-disjunct matrix M 1 V ← the outcome vector

2 D ← ∅

3 T_r ← the set of all r⁰-subsets, r⁰ ≤ r, in N 4 while T_r 6= ∅

5 choose a subset X ∈ T_r and compute t^V₀(X) 6 if t^V₀(X) ≤ e and X 6⊃ X⁰ for all X⁰ ∈ D

7 remove all supersets of X from T_r and D

8 D ← D ∪ {X}

9 else Tr ← Tr− X

10 return D

In general, it is not easy to construct a matrix with error-tolerance. A trivial, but not efficient, construction to obtain error-tolerance is by taking copies of each row of the original matrix. Chen, Du and Hwang [10] extended the substitution-type construction mentioned above to the error-tolerant version. Let Q_z be constructed similar to Q except there are drm + z rows for z ≥ 1. Surprisingly, by replacing Q with Q_z and a (d, r]-disjunct matrix with a (d, r; z⁰]-disjunct matrix respectively in the substitution-type construction, we obtain a (d, r; zz⁰]-disjunct matrix, which can correct up to

¹(zz⁰− 1) 2

º

errors.

Lemma 5.2.4. For any d + 1 edges e0, e1, · · · , ed, there exists a set R of at least z rows in Q_z such that for each j ∈ R none of f_j(e_i) is contained in f_j(e₀), where 1 ≤ i ≤ d.

Proof. By the construction of Q_z, each column is represented by a degree-m poly-nomial in GF (q), and all of which are distinct. Hence any two columns have common entries in at most m rows.

Suppose to the contrary that there exist at most z−1 rows satisfying the condition.

Then by the pigeonhole principle there exists an edge e_x ∈ {e₁, e₂, · · · , e_d} such that there exists a set N of at least rm + 1 rows satisfying f_j(e_x) ⊆ f_j(e₀) for each j ∈ N.

Using the pigeonhole principle again, there exist two columns, one in e_x and the other in e₀\ e_x, with common entries in at least m + 1 rows, a contradiction.

By applying the substitution-type construction to Q_z, we obtain

Theorem 5.2.5. By replacing each q-ary symbol in Qz with a distinct column of a t⁰× q (d, r; z⁰]-disjunct matrix, there exists a (drm + z) · t⁰ × q^m+1 (d, r; zz⁰]-disjunct matrix M.

Proof. It suffices to prove that M is (d, r; zz⁰]-disjunct. Take a pair of disjoint d-set and r-set of columns, we want to show that there exist zz⁰ rows with 1-entries in the designated r columns and 0-entries in the designated d columns.

After transformation, each row satisfying above condition can generate z⁰ rows each of which has a 1-entry in the designated r columns and a 0-entry in the designated d columns. By Lemma 5.2.4, there exist z rows whose entries in the d columns are all different from the entries in the r columns. Hence there exist zz⁰ rows with a 1-entry in each of the r columns and a 0-entry in each of the d columns.

Denote t(n, d, r; z] as the minimum number of rows required for a (d, r; z]-disjunct matrix with n columns.

Corollary 5.2.6. t(q^m+1, d, r; zz⁰] ≤ min{(drm+z)·t(q, d, r; z⁰], (drm+z⁰)·t(q, d, r; z]}

Proof. The proof follows from Theorem 5.2.5 immediately.