Super-simple (v, 5, 2) directed designs and their smallest defining sets with its application in LDPC codes M. Mohammadnezhad, S. Golalizadeh, M. Boostan, N. Soltankhah ∗ arXiv:2007.13743v2 [math.CO] 12 Jul 2022 Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran Abstract In this paper, we show that for all v ≡ 0, 1 (mod 5) and v ≥ 15, there exists a super-simple (v, 5, 2) directed design, also for these parameters there exists a super-simple (v, 5, 2) directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes. Keywords: Super-simple directed design, Smallest defining set, Trade, Directed group divisible design, LDPC codes, Tanner graph 1 Introduction and preliminaries A group divisible design ( or GDD) is a triple (X, G, B) which satisfies the following properties: 1. G is a partition of a set X into subsets called groups; 2. B is a set of subsets of X called blocks such that a group and a block intersect in at most one point; 3. each pair of points from distinct groups occurs in exactly λ blocks. The group type of GDD is the multiset {|G| : G ∈ G}. We use the notation g1u1 g2u2 · · · gnun to denote ui occurrences of gi for 1 ≤ i ≤ n in the multiset. A GDD with block sizes from a set of positive integers K is called a (K, λ)-GDD. When K = {k}, we simply write (k, λ)-GDD. When λ = 1, we simply write K-GDD. A (K, λ)GDD with group type 1v is called a pairwise balanced design and denoted by PBD(v, K, λ). A (k, λ)-GDD with group type 1v is called a balanced incomplete block design, denoted by (v, k, λ)-BIBD. Some generalizations have been introduced for the concept of designs. Gronau and Mullin [17] for the first time, introduced a new definition of block designs called super-simple block designs. A super-simple (v, k, λ) design is a block design such that any two blocks of the design intersect in at most two points. A simple block design is a block design such that it has no repeated blocks. The existence of super-simple (v, 4, λ) designs have been characterized for 2 ≤ λ ≤ 9 , see [4–8, 10, 17, 26, 29]. Also, the existence of super-simple (v, 5, λ) designs have been characterized for 2 ≤ λ ≤ 5, see [9, 11, 12, 14]. A directed group divisible design (K, λ)-DGDD is a group divisible design in which every block is ordered and each ordered pair formed from distinct elements of different groups occurs in exactly λ blocks. A (k, λ)-DGDD with group type 1v is called a directed balanced incomplete block design and denoted by (v, k, λ)-DBIBD or (v, k, λ)DD. A (K, λ)-DGDD is super-simple if its underlying (K, 2λ)-GDD is super-simple. A transversal design, TD(k, λ, n), is a (k, λ)-GDD of group type nk . When λ = 1, we use the notation TD(k, n). Lemma 1. [1] 1 A TD(q + 1, q) exists, consequently, a TD(k, q) exists for any positive integer k(k ≤ q + 1), where q is a prime power. ∗ Corresponding author: soltan@alzahra.ac.ir, soltankhah.n@gmail.com 1 2 A TD(7, n) exists for all n ≥ 63. A set of blocks which is a subset of a unique (v, k, λ)DD is said to be a defining set of the directed design. A minimal defining set is a defining set, no proper subset of which is a defining set. A smallest defining set, is a defining set with the smallest cardinality. A (v, k, t) directed trade of volume s consists of two disjoint collections T1 and T2 each of s ordered k-tuples of a v-set X called blocks, such that every ordered t-tuple of distinct elements of X is covered by exactly the same number of blocks of T1 as of T2 . Such a directed trade is usually denoted by T = T1 − T2 . In a (v, k, t) directed trade, both collections of blocks cover the same set of elements. This set of elements is called the foundation of the trade. In [23], it has been shown that the t minimum volume of a (v, k, t) directed trade is 2⌊ 2 ⌋ and that directed trades of minimum volume and minimum foundation exist. In this paper we use a special type of a directed trade which is defined as follows. Definition 1. [24] Let T = T1 − T2 be a (v, k, 2) directed trade of volume s with blocks b0 , b1 ,· · · , bs−1 such that each pair of consecutive blocks of T1 (bi , bi+1 i = 0, 1, · · · , s − 1 (mod s)) is a trade of volume 2. Such a trade is called a cyclical trade. (We denote a cyclical trade of volume s by CTs ) If D = (V, B) is a directed design and if T1 ⊂ B, we say that D contains the directed trade T . Defining sets for directed designs are strongly related to trades. This relation is illustrated by the following result. Proposition 1. Let D = (V, B) be a (v, k, λ)DD and let S ⊂ B, then S is a defining set of D if and only if S contains a block of every (v, k, 2) directed trade T = T1 − T2 such that T is contained in D. Each defining set of a (v, k, λ)DD D, contains at least one block from each trade in D. In particular, if D contains m mutually disjoint directed trades then the smallest defining set of D must contain at least m blocks. If a directed design D contains a cyclical trade of volume s, then each defining set for D must contain at least ⌊ s+1 2 ⌋ blocks of T1 . Some results have been obtained on (v, k, λ)DDs for special k and λ and their defining sets. For example, in [21], it has been proved that if D is a (v, 3, 1)DD, then a defining set of D has at least v2 blocks. In [16], it has been shown that for each admissible value of v, there exists a simple (v, 3, 1)DD whose smallest defining sets have at least a half of the blocks. In [24], it has been shown that the necessary and sufficient condition for the existence of a super-simple (v, 4, 1)DD is v ≡ 1 (mod 3) and for these values of v except v = 7, there exists a super-simple (v, 4, 1)DD whose smallest defining sets have at least a half of the blocks. Also, in [25], it has been shown that for all v ≡ 1, 5 (mod 10) except v = 5, 15, there exists a super-simple (v, 5, 1)DD such that their smallest defining sets have at least a half of the blocks. In [3], the authors showed that for all v ≡ 1 (mod 3), there exists a super-simple (v, 4, 2)DD such that their smallest defining sets have at least a half of the blocks. In this paper, we prove that the necessary and sufficient condition for the existence of a super-simple (v, 5, 2)DD is v ≡ 0, 1 (mod 5) (v ≥ 15) and for these values of v, there exists a super-simple (v, 5, 2)DD whose smallest defining sets have at least a half of the blocks. We introduce the following quantity d= the total number of blocks in a smallest def ining set in D the total number of blocks in D and we show for all admissible values of v, d ≥ 21 . In the last section we provide a new method to construct parity-check matrices of LDPC codes by these designs. 2 Recursive Constructions For some values of v, the existence of a super-simple (v, 5, 2)DD will be proved by recursive constructions that which are presented in this section for later use. Construction DGDD with index λ1 and with d ≥ 21 . Let S 1. (Weighting)[3] Let (X, G, B) be a super-simple + + w:X →Z {0} be a weight function on X, where Z is the set of positive integers. Suppose that for each block B ∈ B, there exists a super-simpleP(k, λ2 )-DGDD of type {w(x) : x ∈ B} with d ≥ 21 . Then there exists a super-simple (k, λ1 λ2 )-DGDD of type { x∈Gi w(x) : Gi ∈ G} with d ≥ 21 . 2 Construction 2. [3] If there exist a super-simple (k, λ)-DGDD of type g1u1 · · · gtut with d ≥ 12 and a super-simple Pt (gi + η, k, λ)DD for each i(1 ≤ i ≤ t) with d ≥ 21 , then there exists a super-simple ( i=1 gi ui + η, k, λ)DD with d ≥ 21 , where η = 0 or 1. 3 Direct Construction In this section, we construct some super-simple (v, 5, 2)DDs for some small admissible values of v and some super-simple directed group divisible designs by direct construction and for these values of v, we show that the parameter d for constructed designs is at least 21 . In what follows we use the notation +d (mod v), which denotes that all elements of the base blocks should be developed cyclically by adding d (mod v) to them, while the infinite point ∞, if it occurs in the base blocks, is always fixed.We usually omit +d when d = 1. Let [a, b]0,1 5 be the set of positive integers v such that v ≡ 0, 1 (mod 5) and a ≤ v ≤ b. Lemma 2. There exists a super-simple (v, 5, 2)DD for all v ∈ [15, 86]0,1 5 ∪ {95, 110, 111, 115, 116, 130, 131}, whose smallest defining sets have at least a half of the blocks. Proof. For v = 15 and G = Z14 ∪ {∞}, The following base blocks by +2 (mod 14) form a super-simple (15, 5, 2)DD. (1,0,2,3,8) (0,7,11,4,2) (0,3,13,11,9) (1,0,∞,5,7) (0,1,4,10,9) (13,2,∞,0,10) This design contains 42 blocks, each of three columns has 7 disjoint directed trades of volume 2. Since each defining set for this design must contain one 5-tuple of each directed trade in each of columns, then each defining set contains at least 7 × 3 = 21 blocks. So d ≥ 12 . For v = 25 and G = Z24 ∪ {∞}, the following base blocks by +1 (mod 24) form a super-simple (25, 5, 2)DD. (0,5,1,7,15) (2,0,∞ ,17,21) (22,0,5,21,11) (13,6,1,0,9) (12,0,1,10,4) There are 120 blocks in a super-simple (25, 5, 2)DD. The first two columns have 48 disjoint directed trades of volume 2, and the last column is a cyclical trade of volume 24. Since each defining set for this super-simple directed design must contain at least one 5-tuple of each directed trade in the first two columns and 12 5-tuples of cyclical trade in the last column, then each defining set must contain at least 48 + 12 = 60 blocks. Therefore for this super-simple (25, 5, 2)DD the inequality d ≥ 21 is satisfied. For v ∈ [16, 36]0,1 5 except v = 15, 25, the results are summarized in the following table. 3 v 16 20 21 26 30 31 35 36 base blocks (3,0,1,8,6) (7,0,3,11,5) (0,4,3,9,16) (5,0,∞, 7,8) (3,0,6,8,7) (8,2,4,0,16) (5,13,0,7,22) (16,0,11,20,19) (0,3,16,21,23) (1,0,15,2,26) (3,7,0,15,1) (18,0,27,16,26) (6,7,0,30,12) (0,32,11,29,20) (1,0,6,9,21) (4,5,34,0,16) (0,17,3,10,34) (1,7,14,2,11) (0,2,12,7,8) (9,0,1,18,14) (11,2,4,8,0) (0,19,18,7,10) (0,9,14,4,15) (3,0,19,7,25) (0,12,1,24,3) (2,20,10,0,25) (9,0,4,12,26) (9,0,21,3,14) (0,19,9,6,26) (0,24,18,10,31) (5,8,12,20,0) (3,13,0,2,27) (10,14,0,11,30) mod +2 mod 16 bv 48 d≥ mod 19 76 2×19 76 mod 21 84 2×21 84 (8,0,13,14,24) mod 26 130 2×26+13 130 (0,27,4,10,11) (20,8,∞, 1,0) (9,0,27,1,11) (27,0,3,19,2) (0,4,6,5,21) (1,15,∞, 4,0) (13,22,0,18,26) mod 29 174 3×29 174 mod 31 186 3×31 186 (23,0,32,8,25) mod 34 238 3×34+17 238 (2,17,9,30,0) mod 36 252 2×36+36+2×18 252 (2,5,0,1,4) (1,0,10,7,9) 3×8 48 The above table has five columns. The first column contains the values of v and the second column contains the base blocks. The third column shows that how to develope the elements of base blocks. Two last columns contain the number of blocks of corresponding design and the least possible value of d, respectively. For the remaining values of v, their associated super-simple directed designs are presented in the Appendix. Lemma 3. There exists a super-simple (5, 2)-DGDD of type 55 with d ≥ 21 . Proof. Let X = Z25 and let G = {{i, 5 + i, 10 + i, 15 + i, 20 + i}| 0 ≤ i ≤ 4}. Here are the base blocks. These blocks are developed by +5 (mod 25). (4,0,22,21,23) (10,14,3,21,22) (1,18,22,24,0) (16,12,0,23,24) (11,2,9,0,23) (9,3,12,21,15) (2,0,8,6,24) (14,0,2,18,16) (9,7,18,0,6) (0,7,14,21,13) (1,23,14,2,10) (3,6,20,17,9) (4,6,10,13,12) (7,8,0,19,16) (0,7,1,4,3) (2,6,10,18,19) (0,9,13,1,12) (8,4,5,22,16) (3,6,12,14,0) (6,5,3,24,22) This directed group divisible design has 100 blocks, contains 10 disjoint directed trades of volume 2 in each of five columns. Since each defining set for this design must contain one block of each directed trades, then each defining set contains at least 50 blocks. So d ≥ 21 . Lemma 4. For each t, 6 ≤ t ≤ 10, there exists a super-simple (5, 2)-DGDD of type 5t with d ≥ 12 . Proof. Let the point set be Z5t and let the group set be {{i, i + t, i + 2t, i + 3t, i + 4t}| 0 ≤ i ≤ t − 1}. The required base blocks are listed below. All the base blocks are developed by mod 5t. 4 t 6 7 8 9 10 base blocks (2,10,7,17,0) (0,13,16,17,9) (0,1,31,33,30) (11,0,15,20,23) (6,2,28,0,13) (2,23,0,4,1) (16,17,0,30,32) (11,10,21,33,0) (12,21,0,33,25) (18,2,23,45,0) (20,0,19,21,28) (0,2,16,27,19) (19,0,10,25,27) (9,0,19,22,18) (3,0,31,17,10) (22,0,5,25,28) (0,3,20,41,4) (0,2,25,22,41) (12,3,11,0,26) (16,0,11,17,35) (0,16,4,22,33) (0,1,20,12,25) (5,1,10,0,11) (0,12,27,2,31) (11,0,26,37,32) (0,6,31,39,44) (0,13,17,36,28) (2,5,38,0,46) d≥ bt 150 2×30+30 150 210 3×35 210 (11,6,0,20,33) 280 3×40+20 280 (2,0,33,30,37) (0,19,14,17,24) (0,3,31,47,49) (21,28,0,2,37) 360 4×45 360 450 4×50+25 450 (0,21,11,26,25) (0,6,43,7,32) Lemma 5. There exists a super-simple (5, 2)-DGDD of type (15)t for t ∈ {6, 7, 9} with d ≥ 12 . Proof. Let the point set be Z15t and let the group set be {{i, i + t, i + 2t, · · · , i + 14t}| 0 ≤ i ≤ t − 1}. The base blocks are listed below. Here, all the base blocks are developed by mod 15t . t 6 7 9 4 base blocks (32,15,0,83,1) (0,22,68,85,33) (16,56,15,0,43) (40,87,37,59,0) (85,2,88,81,0) (26,51,55,64,0) (0,77,1,57,82) (63,55,0,76,23) (44,0,41,7,15) (74,25,69,16,0) (0,31,57,75,76) (0,11,85,10,100) (0,31,47,51,70) (0,71,43,33,10) (0,50,55,102,72) (15,73,82,0,95) (0,45,73,71,44) (38,0,15,17,49) (27,40,93,0,99) (0,3,55,88,79) (0,80,10,55,69) (0,66,74,82,93) (81,0,62,101,64) (0,34,81,73,99) (76,22,0,58,68) (0,82,31,102,37) (0,51,92,54,94) (19,76,0,44,1) (45,0,43,81,93) (5,0,53,94,69) (0,69,92,96,101) (22,26,60,0,59) (0,46,71,44,61) (25,55,60,0,85) (23,0,105,65,76) (12,49,52,80,0) (38,6,0,107,22) (0,4,2,10,48) (8,4,20,0,96) (22,7,0,51,110) (23,0,84,71,58) (46,0,7,116,85) (70,0,28,59,49) (3,20,0,13,46) (0,11,32,134,66) (88,28,69,35,0) (8,40,57,16,0) (33,52,0,120,91) (6,0,40,92,26) (0,24,98,104,25) (41,3,56,0,70) (46,123,0,125,61) (98,0,74,23,87) (102,44,85,0,14) (97,35,92,14,0) (1,2,0,5,24) (5,0,56,98,118) d≥ bt 1350 7×90+45 1350 1890 9×105 1890 3240 12×135 3240 Main Theorem In this section we try to find super-simple (v, 5, 2)DDs for some admissible values of v by recursive constructions presented in Section 2 and using super-simple DGDDs obtained in Section 3. Lemma 6. There exists a super-simple (v, 5, 2)DD for each v ∈ {20i + η| 5 ≤ i ≤ 9, η = 0, 1} with d ≥ 12 . 5 Proof. Using a super-simple (5, 2)-DGDD of type 5t for 5 ≤ t ≤ 9 with d ≥ 21 obtained in Lemmas 3 and 4 and applying Construction 1 with a TD(5, 4) as an input design comming from Lemma 1, we obtain a supersimple (5, 2)-DGDD of type (20)t with d ≥ 21 . On the other hand by Lemma 2 there exists a super-simple (20 + η, 5, 2)DD. So by Construction 2 we obtain a super-simple (20t + η, 5, 2)DD with d ≥ 12 , where η = 0 or 1. Lemma 7. There exists a super-simple (v, 5, 2)DD for each v ∈ {125, 126, 145, 146, 150, 151} with d ≥ 12 . Proof. We delete 5− a points from the last group of a TD(6, 5) coming from Lemma 1 to obtain a {5, 6}-GDD of type 55 a1 . Applying Construction 1 and using a super-simple (5, 2)-DGDD of group type 55 and 56 with d ≥ 12 from Lemmas 3 and 4 we get a super-simple (5, 2)-DGDD of type (25)5 (5a)1 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (25 + η, 5, 2)DD and a super-simple (5a + η, 5, 2)DD for a ∈ {0, 4, 5} and η = 0, 1, by Construction 2 we get a super-simple (125 + 5a + η, 5, 2)DD with d ≥ 12 . Lemma 8. There exists a super-simple (v, 5, 2)DD for v ∈ {155, 156} with d ≥ 12 . Proof. Starting from a 5-GDD of type 38 71 (exists by Lemma 4.3 in [12]) and applying Construction 1 by using a super-simple (5, 2)-DGDD of type 55 with d ≥ 12 as an input designs, we get a super-simple (5, 2)-DGDD of type (15)8 (35)1 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (15 + η, 5, 2)DD and a super-simple (35 + η, 5, 2)DD, by Construction 2 we get a super-simple (155 + η, 5, 2)DD with d ≥ 21 , where η = 0 or 1. Lemma 9. There exists a super-simple (v, 5, 2)DD for each v ∈ {170, 171, 175, 176, 185, 186} with d ≥ 12 . Proof. Starting from a {5, 6, 7, 8}-GDD of type 47 61 (exists by Lemma 4.4 in [12]) and applying Construction 1 by using a super-simple (5, 2)-DGDD of type 55 , 56 , 57 and 58 with d ≥ 12 coming from Lemmas 3 and 4 we get a super-simple (5, 2)-DGDD of type (20)7 (30)1 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (20 + η, 5, 2)DD and a super-simple (30 + η, 5, 2)DD with d ≥ 21 , then by Construction 2 we get a super-simple (170 + η, 5, 2)DD with d ≥ 12 , where η = 0 or 1. Starting from a T D(5, 7) coming from Lemma 1 and applying Construction 1 by using a super-simple (5, 2)DGDD of type 55 with d ≥ 21 coming from Lemma 3 we get a super-simple (5, 2)-DGDD of type (35)5 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (35 + η, 5, 2)DD, by Construction 2 we get a super-simple (175 + η, 5, 2)DD with d ≥ 12 , where η = 0 or 1. Starting from a {5, 6, 7}-GDD of type 56 71 (exists from Lemma 4.4 in [12]) and applying Construction 1 by using a super-simple (5, 2)-DGDD of type 55 , 56 and 57 with d ≥ 21 coming from Lemmas 3 and 4 we get a super-simple (5, 2)-DGDD of type (25)6 (35)1 with d ≥ 12 . Since by Lemma 2 there exists a super-simple (25 + η, 5, 2)DD and a super-simple (35 + η, 5, 2)DD, by Construction 2 we get a super-simple (185 + η, 5, 2)DD with d ≥ 21 , where η = 0 or 1. Lemma 10. There exists a super-simple (v, 5, 2)DD for any v ∈ {90, 91, 105, 106, 135, 136} with d ≥ 21 . Proof. By Lemma 5 there exists a super-simple (5, 2)-DGDD of type (15)t with d ≥ 12 for t ∈ {6, 7, 9}. Since by Lemma 2 there exist a super-simple (15 + η, 5, 2)DD with d ≥ 12 for η = 0, 1, by Construction 2 we get a super-simple (15t + η, 5, 2)DD with d ≥ 21 , where η = 0 or 1. Lemma 11. There exists a super-simple (96, 5, 2)DD with d ≥ 21 . Proof. A super-simple (5, 2)-DGDD of group type 46 is listed as follows. Let X = Z24 and G = {{i, i + 6, 12 + i, 18 + i}| 0 ≤ i ≤ 5}. Below are the required base blocks. All the base blocks are developed by mod 24. (0,2,1,4,11) (13,2,0,16,21) (1,0,5,22,15) (0,1,20,9,16) 6 This super-simple DGDD has 96 blocks, each of two columns has 24 disjoint directed trades of volume 2. Therefore each defining set for this super- simple DGDD contains at least 24 × 2 = 48 blocks. So d ≥ 21 . Starting from this DGDD and applying Construction 1 with a T D(5, 4) coming from Lemma 1 we get a supersimple (5, 2)-DGDD of type (16)6 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (16, 5, 2)DD with d ≥ 21 , by Construction 2 we get a super-simple (96, 5, 2)DD with d ≥ 12 . Lemma 12. There exists a super-simple (v, 5, 2)DD for any v ∈ {165, 166} with d ≥ 21 . Proof. Let the point set be X = Z33 and the group set be G = {{i, i + 11, i + 22}| 0 ≤ i ≤ 10}. Below are the required base blocks. All the base blocks are developed by mod 33. (6,2,0,3,27) (9,15,19,0,29) (10,0,26,2,19) (1,13,20,0,8) (1,0,4,6,5) (2,0,15,30,5) This super-simple DGDD has 198 blocks, each of three columns has 33 disjoint directed trades of volume 2. Therefore each defining set for this super- simple DGDD contains at least 33 × 3 = 99 blocks. So d ≥ 21 . Starting from this DGDD and applying Construction 1 with a T D(5, 5) coming from Lemma 1 we get a supersimple (5, 2)-DGDD of type (15)11 with d ≥ 21 . Since by Lemma 2 there exists a super-simple (15 + η, 5, 2)DD, by Construction 2 we obtain a super-simple (165 + η, 5, 2)DD with d ≥ 12 , where η = 0 or 1. Lemma 13. Suppose that 5 ≤ k ≤ 10 is an integer. Let N (m) ≥ k−2, r = k−5 and let M = {5m, 5a1 , · · · , 5ar }, where ai ∈ [3, m] ∪ {0}, 1 ≤ i ≤ r. If P there exists a super-simple (l + η, 5, 2)DD with d ≥ 12 for each l ∈ M , then r there exists a super-simple (25m + 5 i=1 ai + η, 5, 2)DD with d ≥ 12 , where η = 0 or 1. Proof. By Lemma 4.8 in [12], there exists a {5, 6, ..., k}-GDD of type m5 (a1 )1 (a2 )1 · · · (ar )1 . Starting from this GDD and applying Construction 1 by using a super-simple (5, 2)-DGDDs of type 5t for t ∈ {5, 6, · · · , k} with d ≥ 12 coming from Lemmas 3 and 4 we get a super-simple (5, 2)-DGDD of type (5m)5 (5a1 )1 (5a2 )1 · · · (5ar )1 with d ≥ 12 . Since there Pr exists a super-simple (u + η, 5, 2)DD for any u ∈ M , by Construction 2 we get a super-simple (25m + 5 i=1 ai + η, 5, 2)DD with d ≥ 21 where η = 0 or 1. 1 Lemma 14. There exists a super-simple (v, 5, 2)DD for any v ∈ [190, 1591]0,1 5 with d ≥ 2 . Proof. Applying Lemma 13 with parameters in the following table, we obtain a every v ∈ [190, 1591]0,1 5 . All required T D(k, m) exist by Lemma 1. Pr Pk−5 v = 25m + 5 i=1 ai + η m k i=1 ai 0,1 [190, 281]5 7 8 [3, 21] [285, 451]0,1 9 10 [12, 45] 5 [455, 651]0,1 13 10 [26, 65] 5 [655, 1251]0,1 25 10 [6, 125] 5 [1255, 1591]0,1 36 10 [71, 138] 5 super-simple (v, 5, 2)DD for η {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} Now, we are in a position to conclude the main result. Main Theorem. For all v ≡ 0, 1 (mod 5) and v ≥ 15, there exists a super-simple (v, 5, 2)DD with d ≥ 12 . Proof. The proof is by induction on v. By the above lemmas, the result is true for v ∈ [15, 1591]0,1 5 . Therefore, we assume that v ≥ 1595. We can write v = 25m+5(a1 +a2 )+η, where m ≥ 63, η = 0, 1, {a1 , a2 } ⊂ [3, m]∪{0} and a1 +a2 ∈ [3, 2m]. By induction there exists a super-simple (5m+η, 5, 2)DD and a super-simple (5ai +η, 5, 2)DD, for i = 1, 2 with d ≥ 12 . Since N (m) ≥ 5, we know that there exists a super-simple (v, 5, 2)DD by Lemma 13. 7 5 Constructing of some LDPC codes by using super-simple (v, 5, 2)DDs A low density parity check (LDPC) codes first proposed by Gallager in 1960 in his dissertation after that these codes were ignored about 36 years and then rediscovered by Mackey [15, 20]. An m × n sparse matrix is one that many of its elements are equal to zero. An LDPC code is a linear code with a sparse parity check matrix. A binary code with parity check matrix with dc 1s in each column and dr 1s in each row is a regular (or bi-regular) LDPC code. Otherwise it is an irregular LDPC code. There is a natural association of an m × n binary matrix with a bipartite graph G = (L ∪ R, E), called Tanner graph [27], whose adjacency matrix is the parity check matrix of the code where columns are identified with the left nodes (L) of the bipartite graph (the message nodes) and rows with the right nodes (R) of the graph (check nodes) and E is the set of edges. The girth of the code is the length of the shortest cycle in the Tanner graph G denoted by g. Reducing the parity check matrix of an LDPC code to standard form to allow systematic coding may render it nonsparse, leading to inefficient encoding. Better encoding techniques are given in [22]. Some approaches to construct LDPC codes are algebric-based, protograph-based, and convolutional LDPC codes. Numerous techniques for the construction of LDPC codes have been proposed. These include the original codes of Gallager [15], MacKay codes [20], irregular degree sequence codes, and codes based on combinatorial structures such as finite geometries and designs [2, 13, 18, 28]. In this section, We use super-simple (v, 5, 2) directed designs to obtain parity-check matrices of trade-based LDPC codes, also we use a graph of trade in which its vertices are associated to the blocks of a super-simple directed design and each edge between two blocks shows a trade of volume 2, and then provide an example to show this method in an irregular LDPC code with girth 8. Let V = {0, 1, . . . , v − 1} be a v-set, corresponding to a super-simple (v, 5, 2)DD with n blocks b1 , b2 , . . . , bn we construct a (v2 ) × n matrix A whose columns indices b1 , b2 , . . . , bn and rows indices (xi , xj ), where xi < xj and xi , xj ∈ {0, 1, . . . , v − 1}, as follows: ( 1 if (xi , xj ) ∈ bl which bl appears in a trade, A(xi ,xj )l = 0 o.w. Then by removing all zero rows and columns of A, the resulting matrix denoted by C is considered as the parity-check matrix of trade-based LDPC code, if the number of rows of C is less than or equal to the number of its columns. otherwise, C ⊥ which is the transpose of C, is taken as the parity-check matrix of the code. This parity-check matrix is the adjacency matrix of the Tanner graph. The number of 1s in each rows and columns is at most 2λ and λ(k − 1), respectively. Because any pair (xi , xj ) where xi < xj , occurs in λ blocks and each block of length k contains k − 1 pairs (xi , xj )s with xi < xj , which may appear in λ trades. Main theorem in previous section, shows that for all v ≡ 0, 1 (mod 5) and v ≥ 15, there exists a super-simple (v, 5, 2)DD whose smallest defining sets have at least half of the blocks. Because of this property we can construct parity-check matrices of LDPC codes in which the number of rows and columns is equal to or bigger than the number of blocks. Thus, the length of our proposed LDPC code is at least the number of blocks. In the super-simple (v, 5, 2)DD, since λ = 2, three cases may occur for its directed trades: 1. Two blocks have a common pair in which first block contains (xi , xj ) and the second one contains (xj , xi ), in this case the directed trade is of the form T1 b1 : (xi , xj , . . .) b2 : (xj , xi , . . .) T2 (xj , xi , . . .) (xi , xj , . . .) 2. Three blocks have a common pair in which two blocks contain (xi , xj ) and the third one contains (xj , xi ), or vice versa. In this case we have two directed trades of volume 2 of the form 8 T1 b1 : (xi , xj , . . .) b3 : (xj , xi , . . .) T2 (xj , xi , . . .) (xi , xj , . . .) T1 b2 : (xi , xj , . . .) b3 : (xj , xi , . . .) T2 (xj , xi , . . .) (xi , xj , . . .) the third block that has (xj , xi ) is common in two trades. 3. Four blocks have a common pair in which two blocks contains (xi , xj ) and two other ones contains (xj , xi ), in this case we have four directed trades of volume 2 as shown below: T2 T2 T2 T1 T1 T1 b1 : (xi , xj , . . .) (xj , xi , . . .) b1 : (xi , xj , . . .) (xj , xi , . . .) b2 : (xi , xj , . . .) (xj , xi , . . .) b3 : (xj , xi , . . .) (xi , xj , . . .) b4 : (xj , xi , . . .) (xi , xj , . . .) b3 : (xj , xi , . . .) (xi , xj , . . .) T1 T2 b2 : (xi , xj , . . .) (xj , xi , . . .) b4 : (xj , xi , . . .) (xi , xj , . . .) Next proposition describes the relation between the smallest volume of cyclical trade and the girth of tradebased LDPC code, which is used to construct the codes. Proposition 2. [2] A super-simple directed design has a cyclical trade of volume s if and only if the Tanner graph of the corresponding trade-based LDPC code has 2s-cycles. Proposition 3. The girth of an LDPC code constructed by using super-simple (v, 5, 2)DD is at least 6. Proof. From proposition 2, we conclude that the girth of the Tanner graph of the corresponding trade-based LDPC code constructed by using super-simple (v, 5, 2)DD must be even. By the concept of super-simple directed designs every two blocks intersects in at most two points or an (xi , xj ), therefore the Tanner graph is 4-cycle free, in other words there is no cycle in its Tanner graph of the following type: Bi Bj (yi , yj ) (xi , xj ) If we have cyclical trade of volume 3 results in 6-cycle in the Tanner graph of a trade-based LDPC code, otherwise the code has girth at least 8. Combining proposed technique to construct a parity-check matrix for an LDPC code based on the concept of trades in super-simple directed designs that explained above, proposition 2 and proposition 3, we have the following result. Theorem 1. The existence of a super-simple (v, 5, 2)DD whose smallest defining set have at least half of the blocks can deduce an LDPC code with the girth at least 6. We define a special set that knowledge of this is important for a given LDPC code. We show the graphical structure of this set corresponding to the LDPC code in the next example. Definition 2. [19] Let G be the Tanner graph of an LDPC code given by the null space of a parity-check matrix Cm×n . For 1 ≤ α ≤ n , 1 ≤ β ≤ m, an (α, β)-trapping set is a subset D ⊆ V of variable nodes in the Tanner graph of LDPC code such that |D| = α and |O(D)| = β, where O(D) is the subset of check nodes of odd degree in an induced subgraph of the tanner graph GD . An elementary (α, β)-trapping sets ((α, β) ETSs) are those trapping sets for which all check nodes in the induced subgraph of the Tanner graph is of degree 1 or 2. In an (α, β) ETS, by removing all check nodes of odd degree and replacing every check nodes of even degree with an edge we obtain a graph with α vertices which is called normal graph. There is a 2s-cycle in the ETS if and only if there is a s-cycle in its corresponding normal graph. We use this fact to find easily the girth of the Tanner graph. 9 Figure 1: The graph corresponding to directed trades of super-simple (15, 5, 2) DD Example 1. The followings are the blocks of a super-simple (15, 5, 2)DD in lemma 2 : B1 = (1, 0, 2, 3, 8) B15 = (0, 1, 4, 10, 9) B29 = (1, 0, ∞, 5, 7) B2 = (3, 2, 4, 5, 10) B16 = (3, 2, 6, 12, 11) B30 = (2, 3, ∞, 7, 9) B3 = (5, 4, 6, 7, 12) B4 = (7, 6, 8, 9, 0) B17 = (5, 4, 8, 0, 13) B18 = (7, 6, 10, 2, 1) B31 = (4, 5, ∞, 9, 11) B32 = (6, 7, ∞, 11, 13) B5 = (9, 8, 10, 11, 2) B19 = (9, 8, 12, 4, 3) B33 = (8, 9, ∞, 13, 1) B6 = (11, 10, 12, 13, 4) B7 = (13, 12, 0, 1, 6) B20 = (11, 10, 0, 6, 5) B21 = (13, 12, 2, 8, 7) B34 = (10, 11, ∞, 1, 3) B35 = (12, 13, ∞, 3, 5) B8 = (0, 3, 13, 11, 9) B9 = (2, 5, 1, 13, 11) B22 = (0, 7, 11, 4, 2) B23 = (2, 9, 13, 6, 4) B36 = (13, 2, ∞, 0, 10) B37 = (1, 4, ∞, 2, 12) B10 = (4, 7, 3, 1, 13) B24 = (4, 11, 1, 8, 6) B38 = (3, 6, ∞, 4, 0) B11 = (6, 9, 5, 3, 1) B12 = (8, 11, 7, 5, 3) B25 = (6, 13, 3, 10, 8) B26 = (8, 1, 5, 12, 10) B39 = (5, 8, ∞, 6, 2) B40 = (7, 10, ∞, 8, 4) B13 = (10, 13, 9, 7, 5) B27 = (10, 3, 7, 0, 12) B41 = (9, 12, ∞, 10, 6) B14 = (12, 1, 11, 9, 7) B28 = (12, 5, 9, 2, 0) B42 = (11, 0, ∞, 12, 8) As shown in Fig. 1, each block appeares in {2, 3, 4, 5} trades of volume 2 (We show each Bi by i, i ∈ {1, 2, · · · 42} in Fig. 1). The graph is free of triangles which showes that there is no cyclical trades of volume 3. In the normal graph the smallest cycle is of size 4 which corresponds to a CT4 . By using this design we can construct a matrix C which has the same 42 rows and columns. Therefore C is the parity-check matrix of an irregular LDPC code with dc = {1, 2, 3, 4} ones in columns and dr = {2, 3, 4} ones in rows. Fig. 2 corresponds to the tanner graph of irregular LDPC code of super-simple (15, 5, 2) DD which the girth of it is 8, this girth that shown in Fig. 3 can be g = {B2 , (4, 5), B31 , (9, 11), B14 , (7, 9), B30 , (2, 3), B2 }. The same can be done for other existing super-simple (v, 5, 2)DD. 10 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 B18 B19 B20 B21 0,1 B22 0,2 B23 0,7 B24 0,12 B25 0,∞ B26 1,3 B27 1,5 B28 1,8 B29 1,11 B30 1,13 B31 2,3 B32 2,4 B33 2,9 B34 2,∞ B35 3,5 B36 3,7 B37 3,10 B38 3,13 B39 4,5 B40 4,6 B41 4,11 B42 4,∞ 5,7 5,9 5,12 6,7 6,8 6,13 6,∞ 7,9 7,11 8,9 8,10 8,∞ 9,11 9,1310,1110,1210,∞11,1312,1312,∞ Figure 2: The Tanner graph of irregular LDPC code of girth 8 corresponding to super-simple (15, 5, 2)DD B31 B2 B14 B30 B31 B2 4, 5 B30 9, 11 B14 7, 9 B8 2, 3 B13 Figure 3: A normal graph (4-cycle) and its corresponding (4, 2) ETS (8-cycle) 11 Appendix base blocks (7,8,18,5,0) (0,38,22,12,30) (0,4,1,11,29) (36,39,28,7,0) (0,11,21,2,7) (0,4,10,23,12) (0,11,13,27,36) (0,28,∞, 4,33) (6,8,27,0,32) (1,19,0,25,15) (0,40,9,26,43) (0,41,24,5,36) (0,14,17,2,22) (6,0,1,38,18) (0,11,7,10,23) (0,8,38,36,29) (0,18,38,32,39) (3,9,∞, 0,32) (0,20,24,30,6) (0,7,26,35,3) (0,39,20,6,15) (0,23,1,27,17) (0,3,1,29,19) (0,16,43,14,36) mod 39 bv 312 4×39 312 mod 41 328 4×41 328 mod 44 396 4×44+22 396 46 (22,0,9,33,37) (14,1,7,0,10) (0,7,2,16,33) (12,0,24,26,32) (10,0,15,37,28) (31,0,28,25,29) (17,0,25,35,38) (0,5,24,22,45) (0,20,1,36,31) mod 46 414 4×46+23 414 50 (23,0,30,42,34) (0,15,8,47,48) (0,32,41,11,45) (5,45,0,19,42) (0,30,45,47,16) (1,23,0,3,31) (0,38,35,41,10) (0,5,25,12,36) (20,0,27,33,21) mod 49 490 5×49 490 (5,0,17,27,43) (0,5,∞,14,39) (40,17,0,22,29) (15,0,18,47,31) (17,27,30,32,0) (6,42,0,32,44) (0,33,20,39,50) (0,4,31,41,8) (5,27,35,0,42) (28,2,42,16,0) mod 51 510 5×51 510 (24,30,0,50,47) (7,8,5,50,0) (15,25,53,3,0) (2,20,3,47,0) (6,8,0,28,31) (0,9,5,13,16) (14,7,22,0,40) (0,20,31,45,30) (6,0,21,42,44) (5,11,∞,0,24) mod 54 594 5×54+27 594 (0,40,21,39,41) (4,2,32,0,36) (0,23,15,20,28) (0,51,11,44,26) (0,7,29,6,46) (6,20,13,32,0) mod 56 616 2×56+4×56 616 (4,1,0,10,52) (3,0,1,17,34) (0,10,42,45,53) (0,46,12,47,41) (2,37,5,0,41) (0,3,8,21,32) (6,31,0,19,52) (0,5,16,42,6) (0,44,∞,14,23) (16,0,4,40,31) mod 59 708 6×59 708 (8,0,2,20,45) (0,41,10,40,48) (3,0,55,1,21) (3,0,9,2,43) (25,10,0,32,12) (5,24,0,20,50) (0,2,6,49,42) (6,0,18,4,25) (4,0,12,23,37) (12,0,36,8,50) (24,8,0,46,13) (0,16,48,31,26) mod 61 732 6×61 732 (32,0,35,1,52) (35,0,3,44,34) (10,25,0,23,39) (0,41,∞,31,52) (0,11,24,16,39) (0,48,22,32,17) (23,20,0,22,48) (0,10,22,27,1) (0,23,19,53,51) (0,46,25,19,49) (0,27,36,35,5) (0,37,51,31,57) mod 64 832 6×64+32 832 (0,7,45,16,56) (5,47,7,0,53) (4,49,44,0,56) (9,56,0,10,13) v 40 41 45 51 55 56 60 61 65 (4,13,18,30,0) (5,0,35,22,34) (4,29,0,10,48) (0,9,27,13,30) (9,11,25,0,34) d (17,0,6,33,52) (28,14,0,32,61) 12 v 66 70 71 75 76 80 81 85 base blocks (17,37,36,0,65) (20,0,31,62,24) (15,27,5,0,24) (6,29,63,0,42) (22,4,0,20,27) (5,0,35,57,9) (34,0,6,32,59) (0,3,14,15,55) (0,26,33,52,64) (9,23,38,0,62) (45,0,58,39,47) (29,0,43,35,45) (5,59,0,25,43) (32,0,44,23,57) (32,16,0,33,54) (0,9,11,59,51) (4,27,0,20,28) (8,48,41,35,0) (0,7,4,27,42) (10,0,17,25,43) (3,13,0,49,54) mod 66 bv 858 6×66+33 858 (18,37,47,0,65) (29,33,0,9,50) (30,35,0,37,36) (2,37,∞,13,0) mod 69 966 7×69 966 (22,52,0,55,49) (6,12,0,4,58) (20,0,67,49,52) (47,0,69,19,37) (0,16,13,21,43) (1,0,15,56,59) (0,66,41,1,58) (1,6,0,31,14) (0,40,7,13,48) (23,27,0,20,56) mod 71 994 7×71 994 (43,0,55,34,45) (62,0,16,17,5) (0,30,19,25,47) (0,44,36,51,57) (34,0,62,50,60) (16,25,0,70,11) (5,1,0,73,57) (37,0,4,33,55) (19,0,43,53,21) (0,67,15,47,18) (18,36,0,20,47) (0,36,21,60,61) (40,55,0,8,60) (0,23,31,16,28) mod 74 1110 7×74+37 1110 (0,30,39,10,65) (32,21,64,0,44) (36,17,58,0,71) (0,36,53,20,55) (24,0,72,31,65) (0,4,∞,66,68) (5,58,7,0,65) (0,38,63,37,48) (0,32,1,49,46) (0,3,14,38,64) (0,5,51,55,67) (6,12,0,5,45) mod 76 1140 7×76+38 1140 (0,9,1,67,22) (29,37,52,0,73) (21,38,20,0,28) (37,70,0,20,68) (0,3,4,52,23) (11,45,0,73,56) (32,8,33,0,11) (40,0,61,39,63) (10,0,54,6,52) (0,30,17,26,33) (0,50,44,51,40) (73,0,22,54,20) (42,0,74,30,68) mod 79 1264 8×79 1264 (0,38,∞,64,10) (0,66,41,53,71) (49,0,47,26,30) (0,41,15,49,31) (16,4,7,0,52) (48,45,61,56,0) (27,38,0,65,79) (0,50,7,75,78) (0,2,44,17,76) (0,37,53,67,72) (7,31,66,68,0) (8,0,7,64,18) (65,23,29,0,33) (0,35,62,12,21) (0,16,58,21,52) (37,42,0,74,46) mod 81 1296 8×81 1296 (52,0,17,60,78) (9,18,0,45,57) (4,7,60,0,82) (0,10,45,15,75) (14,0,20,68,69) (28,48,0,51,70) (48,0,68,25,11) (17,35,0,48,73) (0,33,45,67,73) (57,0,59,80,76) (25,41,0,57,58) (15,0,41,67,74) (0,56,40,69,70) (9,34,19,4,0) (0,18,68,30,47) (56,38,0,53,42) mod 84 1428 8×84+42 1428 (18,52,57,0,55) (5,0,∞,19,40) (1,22,0,72,65) (75,0,76,4,83) (4,0,78,28,48) (0,49,72,51,57) (29,0,60,38,40) (3,26,48,40,0) (0,29,71,39,6) 13 d (12,25,28,65,0) (15,0,27,42,56) (27,0,8,37,51) (0,49,64,4,40) (19,0,43,65,13) v 86 95 110 111 115 116 base blocks (68,41,72,69,0) (13,64,0,12,67) (0,59,84,31,33) (10,0,72,70,82) (0,63,43,7,78) (0,23,41,50,47) (0,22,77,21,79) (0,27,5,37,46) (30,78,24,0,44) (16,0,42,65,19) (0,36,26,78,65) (0,8,69,82,75) (79,0,38,74,49) (0,39,1,17,54) (17,46,70,0,28) (0,20,77,66,71) (5,18,39,0,43) (3,77,0,1,36) (49,53,0,77,84) (40,1,0,13,4) (21,73,0,81,38) (0,80,82,64,69) (79,0,89,14,10) (48,0,92,67,59) (46,7,0,48,75) (3,68,86,52,0) (0,6,∞,20,72) (12,28,62,0,13) (55,0,26,88,37) (19,58,0,63,25) (0,31,88,16,40) (38,72,1,8,0) (23,0,3,50,43) (32,0,28,23,70) (0,101,30,34,43) (0,18,26,40,98) (36,0,103,19,83) (40,85,0,88,97) (46,0,74,57,98) (40,38,0,31,48) (0,1,44,50,67) (4,70,81,14,0) (0,100,73,78,85) (23,48,63,0,104) (0,33,87,46,53) (65,8,0,55,84) (0,15,108,94,75) (3,19,∞,54,0) (0,59,77,79,81) (86,37,0,107,102) (0,74,71,108,61) (0,36,50,85,32) (8,0,80,85,104) (2,19,50,41,0) (21,0,86,59,89) (0,6,33,97,27) (33,69,68,0,76) (0,68,5,105,79) (51,26,0,96,80) (0,32,1,37,63) (0,55,87,97,57) (57,47,0,88,103) (0,33,34,108,86) (44,28,104,0,110) (0,3,20,107,86) (53,36,56,0,80) (51,0,63,91,64) (5,0,51,107,41) (23,0,52,96,39) (0,59,15,40,77) (79,6,27,77,0) (38,83,0,61,65) (11,0,93,79,33) (0,47,49,58,72) (0,106,11,1,109) (18,50,106,87,0) (0,69,81,95,99) (17,12,0,38,47) (0,113,53,111,102) (0,4,6,83,29) (33,20,82,0,90) (0,53,50,63,69) (46,16,0,74,1) (80,98,57,112,0) (50,43,30,0,42) (93,100,34,79,0) (0,102,12,79,82) (51,101,34,0,30) (0,85,11,74,7) (88,10,26,0,51) (75,97,68,101,0) (54,0,71,18,5) (42,0,62,108,15) (36,0,76,44,86) (0,100,24,51,29) (92,0,9,37,75) (1,94,67,88,0) (8,0,42,65,11) (0,75,∞,31,6) (84,75,0,56,23) (0,101,63,54,68) (41,56,0,87,113) (65,41,0,10,109) (0,52,72,19,34) (11,16,0,35,67) (99,5,0,16,12) (2,0,92,35,89) (0,62,55,99,100) (0,64,71,18,74) (26,0,41,114,107) (114,0,50,82,108) (0,37,61,25,23) (24,0,89,99,103) (66,68,95,74,0) 14 (63,0,106,39,115) (0,35,41,20,60) (0,4,34,102,74) (34,0,12,98,9) (0,47,77,90,85) (19,0,55,86,73) (12,0,106,51,71) (40,20,99,36,0) (33,69,0,1,46) (39,0,28,97,72) mod 86 bv 1462 8×86+43 1462 d mod 94 1598 8×94+47 1598 mod 109 2398 11×109 2398 mod 111 2442 11×111 2442 mod 114 2622 11×114+57 2622 mod 116 2668 (6+2+4)×116 2668 v 130 131 base blocks (26,67,0,16,97) (101,36,0,95,79) (100,18,0,19,91) (0,82,11,21,34) (71,12,0,75,127) (0,12,49,110,63) (0,45,18,82,110) (57,122,79,0,120) (0,54,57,59,108) (3,0,112,70,69) (0,83,114,11,123) (0,124,41,101,74) (11,0,105,111,71) (0,24,32,46,122) (8,26,∞,51,0) (40,52,36,0,56) (1,32,33,0,116) (10,0,101,7,20) (0,26,50,94,53) (23,11,0,85,127) (0,39,68,77,35) (0,15,105,42,122) (0,69,39,75,92) (0,30,5,38,114) (95,0,21,40,120) (8,93,0,108,52) (0,70,1,17,59) (70,53,11,69,0) (0,118,2,34,9) (32,20,0,13,54) (68,0,4,18,105) (26,49,0,117,93) (8,36,5,0,79) (55,0,52,98,103) (16,0,72,10,27) (72,62,45,56,0) (0,40,108,26,64) (18,81,0,14,44) (104,125,39,0,29) (100,71,92,0,35) (78,119,58,77,0) (77,130,89,0,19) (0,25,107,23,116) (129,23,47,0,38) (101,50,0,46,83) (127,76,46,0,94) (22,9,0,106,7) (0,124,13,112,90) (0,28,36,88,31) (128,80,52,0,85) mod 129 bv 3354 13×129 3354 d mod 131 3406 13×131 3406 (123,21,92,0,57) (0,104,65,110,75) References [1] R.J.R. 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