LDPC codes from triangle-free line sets Keith E. Mellinger
KEITH. E. MELLINGER Department of Mathematics Mary Washington College 1301 College Avenue, Trinkle Hall Fredericksburg, VA 22401
[email protected] Keywords LDPC codes, cap, polarity Abstract We study sets of lines of AG(n, q) and P G(n, q) with the property that no three lines form a triangle. As a result the associated point-line incidence graph contains no 6-cycles and necessarily has girth at least 8. One can then use the associated incidence matrices to form binary linear codes which can be considered as LDPC codes. The relatively high girth allows for efficient implementation of these codes. We give two general constructions for such triangle-free line sets and give the parameters for the associated codes when q is small.
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Introduction
Let AG(n, q) and P G(n, q) be the classical finite affine and projective spaces respectively of dimension n over the finite field GF (q) and let L be a subset of lines of one of these spaces. We say that L is triangle-free if no three nonconcurrent lines of L intersect pairwise and hence form a triangle. The motivation for studying such line sets comes from their connection to a problem in graph theory and coding theory. Let G be the bipartite graph whose first partition set is the set of points of some finite space and whose second partition set contains the lines of such a triangle-free set L of the same space. Edges are naturally determined by incidence. Since L is triangle-free, G contains no cycles of length 6 and hence has girth at least 8. The associated incidence matrix for G is relatively sparse and so can be used to construct a low-density parity check (LDPC) code as originally introduced by Gallager [3]. We let the graph G be the so-called Tanner graph for the linear code C as described in [10]. LDPC codes have become increasingly popular due to their performance in iterative decoding algorithms which approaches the Shannon limit [8] under certain conditions. Tanner’s graphical representation of LDPC codes [10] influenced much of the current literature. The principal method of designing such codes is somewhat random, and explicit constructions are needed for implementation purposes as well as for understanding the properties of these codes. Some examples of LDPC codes constructed from a family of graphs with relatively high
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girth can be found in [6] and other constructions using finite geometry can be found in [7]. We give an explicit construction of such codes based on triangle-free line sets.
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Bounds on triangle-free line sets
We start by looking at sets of lines of AG(2, q) such that no three form a triangle. Suppose the three lines l1 , l2 , and l3 form a triangle. Then the three lines must certainly come from three distinct parallel classes. This give us the following straightforward result. Lemma 2.1 Let L be an arbitrary set of lines of AG(2, q). If L is triangle-free, then |L| ≤ 2q. Proof. If all of the lines of L were from at most two distinct parallel classes, then the result follows immediately. Now suppose the lines of L fall into more than two distinct parallel classes. Let l1 , l2 , and l3 be three lines of L from distinct parallel classes. Then, since any pair of these three lines must meet, they necessarily form a triangle, unless the lines are concurrent. By induction, the only possibility is that all lines are concurrent. Since the number of lines through a point of AG(2, q) is q + 1, the result follows. ¥ Example 2.2 Let L be the union of lines from two distinct parallel classes of AG(2, q). Then clearly L is triangle-free since any set of three lines will contain at least two which are parallel. Moreover, any additional line will necessarily meet every line of L, hence forming a triangle. Therefore, L is maximal. Example 2.3 Let L be the set of all q + 1 lines through a given point. Then, since q + 1 ≥ 3, any additional line will necessarily meet at least two lines of L. Hence, L is triangle-free and maximal. We use the term maximal to mean that the set cannot be extended to any larger triangle-free set. We now examine triangle-free line sets in higher dimensional affine spaces. Let L be a triangle-free set of lines in AG(n, q). Lemma 2.4 If l is a line of L, then the number of distinct lines m 6= l of L which meet l in a point is at most q(q n−2 + · · · + q + 1). Proof. The number of planes through the line l is exactly q n−2 + · · · + q + 1. From the proof of Lemma 2.1, each of these planes can contain at most q additional lines from the set L which meet l. This gives us the count in the lemma. ¥ Theorem 2.5 If L is a triangle-free line set of AG(n, q), then |L| ≤ q n−1 (q n−2 + · · · + q + 2). Proof. We use a technique known as quadratic counting to count the set of triples (l1 , l2 , p) where l1 and l2 are distinct lines of L which intersect in the point p. Let mp represent thePnumber of lines of L which pass through the point p, and let M = |L|. Then p mp = M q, and by counting the points p first, the number of desired triples is X X (m2p − mp ). [mp (mp − 1)] = p
p
LDPC codes from triangle-free line sets
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P
m2p is minimized precisely when all values of mp are P 2 equal. In this case, mp = and hence p m2p = qM n−2 . Therefore, the number of triples (l1 , l2 , p) as defined above is at least We now use the fact that
p Mq qn
M2 − M q. q n−2 On the other hand, counting l1 first gives us M choices for l1 . Then, from the lemma, the number of choices for l2 is at most q(q n−2 + · · · + q + 1). This uniquely determines p. Hence, the number of triples (l1 , l2 , p) is at most M q(q n−2 +· · ·+q+1). Putting these two counts together, M2 − M q ≤ M q(q n−2 + · · · + q + 1) q n−2 which implies
M ≤ q n−1 (q n−2 + · · · + q + 2). ¥
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A construction using caps
We can construct a family of triangle-free line sets using caps. Recall that a cap is a collection of points, no three collinear. Let Σ represent the projective space P G(n, q) and let H∞ be a hyperplane of Σ so that Σ∗ = Σ \ H∞ models the affine space AG(n, q). Let C be a cap in the hyperplane H∞ and consider the set of all lines LC of Σ∗ which meet H∞ in a point of C. Proposition 3.1 The set of lines LC forms a triangle-free line set of AG(n, q). Proof. For contradiction, suppose that there are three lines l1 , l2 , and l3 of LC which form a triangle. Then in the projective completion of Σ∗ these three coplanar lines meet the hyperplane H∞ in three distinct points of C which are necessarily collinear contradicting the cap property. ¥ Note that the bound of Theorem 2.5 is obtained for n = 3. That is, in AG(3, q), q even, a maximal triangle-free line set is obtained from the construction above using a hyperoval in the hyperplane at infinity. We discuss this construction at the end of this section. When n = 4, the maximal size for a cap in P G(3, q) is q 2 + 1 for all q. Hence, the set of lines LC formed from this construction has cardinality q 3 (q 2 + 1), and the bound of q 3 (q 2 + q + 2) is not met. We use the triangle-free line sets of AG(3, q) as defined above to generate LDPC codes. We start by letting q be an odd prime power. In this case, recall that largest possible cap in P G(2, q) is called an oval. Moreover, when q is odd, every oval contains q + 1 points and can be realized as the set of points satisfying some nondegenerate quadratic form on the coordinates, a conic [9]. We fix H∞ to be the plane given by all points whose first homogeneous coordinate is 0. Since all ovals are conics (when q is odd), and all non-degenerate conics of P G(2, q) are equivalent [4], without loss of generality, we fix the conic C defined by the points {(0, 1, x, x2 ) : x ∈ GF (q)} ∪ {(0, 0, 0, 1)}.
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Let MC be the associated q 2 (q + 1) × q 3 incidence matrix over the finite field GF (2). We label the columns of MC with the points of AG(3, q) and the rows of MC with the lines of our triangle-free line set defined in the previous section and using the conic C given above. Example 3.2 Let q = 3. Then, the conic C contains the 4 points (0, 0, 0, 1), (0, 1, 0, 0), (0, 1, 1, 1), and (0, 1, 2, 1). There are exactly q 2 (q + 1) = 36 lines of AG(3, 3) which meet H∞ in a point of C. Moreover, every point of C has q 2 = 9 affine lines through it. Generating the desired lines amounts to finding the 2dimensional vector subspaces (not representing lines contained in H∞ ) which pass through the vectors above, and then normalizing the (non-zero) vectors in these subspaces to represent the 3 distinct affine points on each line. These lines can be enumerated as follows: point p of C (0, 0, 0, 1) (0, 1, 0, 0) (0, 1, 1, 1) (0, 1, 2, 1)
lines l(x,y) l(x,y) l(x,y) l(x,y)
of AG(3, 3) through p, for x, y ∈ GF (3) = {(1, x, y, k) : k ∈ GF (3)} = {(1, k, x, y) : k ∈ GF (3)} = {(1, k, x + k, y + k) : k ∈ GF (3)} = {(1, k, x + 2k, y + k) : k ∈ GF (3)}
The incidence matrix in this case is a 36 × 27 matrix. Each row has weight 3 (the number of points on a line) and each column has weight 4 (the number of lines through a point). Let GC be the bipartite graph determined from the incidence matrix MC and let C be the code with parity check matrix MC . Similarly, let C T represent the code with parity check matrix MCT . We have the following results about our graphs and codes. Proposition 3.3 The graphs GC have girth 8. Proof. Since our line set LC contains no triangles, note that the girth must be greater than 6. We show the girth is 8 by exhibiting an 8-cycle. We represent points using homogeneous coordinates in the projective space, and lines are represented with equations on the homogeneous coordinates (w, x, y, z). Our desired 8-cycle is given by (1, 0, 0, 0) → {y = z = 0} → (1, 1, 0, 0) → {w = x, y = 0} → (1, 1, 0, 1) → {w = z, y = 0} → (1, 0, 0, 1) → {x = y = 0} → (1, 0, 0, 0). ¥ Theorem 3.4 The minimum distance of the codes C generated from GC is at least 2(q + 1). The minimum distance of the codes C T is between 2q and 2q + 2. Proof. We know that each point lies on q + 1 lines of LC and each line contains q points. The lower bounds now follow from [10, Theorem 2] since we know that the girth of GC is 8. We find the upper bound for the minimum distance of C T using geometric properties. Let R be a hyperbolic quadric of Σ which intersects the plane H∞ in exactly the conic C [5]. Certainly the quadric R is not unique. Now, R is ruled by two families of q + 1 lines each. These two ruling families are commonly called reguli. Each regulus forms a partition of the points of R, and so every point of R lies on exactly one ruling line from each of the reguli. Moreover, since R meets H∞ in the conic C, every ruling line of R must meet H in a point of C.
LDPC codes from triangle-free line sets
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Table 1: Codes from caps with q = 3, 4, 5 q C CT
3 [27,6,12] [36,15,6]
4 [64,19,16] [80,35,8]
5 [125,40,20] [150,65,10]
Table 2: Codes from caps with q = 7, 8, 9, 11 q C CT
7 [343,126] [392,175]
8 [512,223] [576,287]
9 [729,288] [810,369]
11 [1331,550] [1452,671]
Let LR be the set of affine lines corresponding to the ruling lines of R. Clearly LR is a subset of LC . Let v be the characteristic vector corresponding to the line set LR . Then, since every point of AG(3, q) is lies on either 0 or 2 lines of LR , the vector v is orthogonal to every row of the parity check matrix MCT . Therefore, the code C T contains a vector of weight |LR | = 2(q + 1) which proves the statement. ¥ Using the software package Magma [2], one can compute the parameters for some of the codes C and C T . The parameters for these codes are summarized in Tables 1 and 2. Here, we note that the [27, 6, 12] code is optimal [1]. Based on the data from the tables, we can make a conjecture as to the dimension of these codes when q is odd. Conjecture 3.5 For q odd, the codes C defined above are [q 3 , q(q − 1)2 /2]-codes. The codes C T are [q 2 (q + 1), q(q 2 + 1)/2]-codes. Note that if this conjecture is true, the information rate for these codes approaches 1/2. Because of the geometry of the triangle-free line sets, one can comment on the automorphism group of these codes. Proposition 3.6 The automorphism group of the code C contains a subgroup of order q 4 (q − 1)(q 2 − 1). Proof. The group fixing the space H∞ pointwise has order q 3 (q − 1). This is the group of translations of AG(3, q) [4] and is a normal subgroup of Aut(P G(3, q)). In addition, the automorphism group of the non-degenerate conic C, isomorphic to P GO(3, q), has order q(q 2 − 1). These groups clearly meet in only the identity. Their semidirect product has the desired order. Since both of these groups fix the triangle-free line set LC , the result follows. ¥ When q is even, one can construct similar codes using hyperovals in the hyperplane at infinity. Recall that when q is even, the maximum number of points in
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Table 3: Codes from Hyperovals q C CT
2 [8,1,8] [16,9,4]
4 [64,18,16] [96,50,8]
8 [512,213] [640,341]
P G(2, q), no three collinear, increases from q + 1 to q + 2. Such a set of points is called a hyperoval. If C is a conic of P G(2, q), q even, then the tangent lines to C all meet in a common point called the nucleus (or knot) and the points of C together with the nucleus form a hyperoval. Using the conic defined above, the points of the associated hyperoval are given by {(0, 1, x, x2 ) : x ∈ GF (q)} ∪ {(0, 0, 0, 1)} ∪ {(0, 0, 1, 0)}. Here, the point (0, 0, 1, 0) is the nucleus. Hyperovals constructed from conics are called regular and every hyperoval in P G(2, q), when q = 2,4, or 8, is regular [4]. As a result, all codes generated by hyperovals when q = 2,4, or 8 are equivalent. They are described in Table 3. Computing the parameters for these codes with q ≥ 24 is unfortunately out of reach with today’s technology. It is certainly conceivable that codes constructed from non-regular hyperovals could have different parameters from the codes constructed using a regular hyperoval of the same space. We note that the [16, 9, 4] code is optimal [1]. A code with the same parameters (again with associated girth 8) was also constructed from an algebraically defined graph in [6]. As before, [10, Theorem 2] give us the following result on the minimum distance. Theorem 3.7 The minimum distance of the codes C generated from hyperovals is at least 2(q + 2). The minimum distance of the codes C T is at least 2q. The diameter of the associated graphs GC also plays a role in the analysis of the codes. For our codes, we can compute the diameter exactly. Proposition 3.8 The diameter of GC is either 4 or 6 depending on whether C is an oval or hyperoval, respectively. Proof. Let P and Q be two distinct points of AG(3, q), and let l be the line determined by P and Q. If l is a line of LC , then the distance between the vertices representing P and Q in GC is 2. If l is not a line of LC , then there exists a plane through P and Q meeting the hyperplane H∞ in a line that intersects the cap C in two distinct points, say R and S. A simple counting argument shows that such a plane exists as long as C is maximal. Then the lines P R and QS are coplanar and so meet in an affine point T . The point line sequence P , P R, T , QS, Q gives a path of length 4 between the vertices represented by P and Q in the graph GC . Now let P be a point of AG(3, q) and let l be a line of LC . If P is on l, then the vertices labelled with P and l in GC are adjacent. If not, then let B be the unique point of C that lies on l. The sequence P , P B, B, l gives a path of length 3 between P and l.
LDPC codes from triangle-free line sets
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Now suppose l and m are two lines of LC . If l and m share an affine point, we are done. So suppose they do not meet in an affine point. Let A and B be the two points of C which lie on l and m respectively, and first suppose A 6= B. There are q 2 distinct affine lines, not necessarily in LC , that intersect both l and m in affine points. Since l and m are not coplanar, none of these lines can meet the plane containing C in a point of the line AB. Moreover, no two of these lines can meet the plane containing C in the same point. It follows that some line, say l∗ , that intersects both l and m in an affine point must meet the conic C and is therefore in LC . Suppose l∗ meets l and m in points X and Y respectively. Then, the point-line sequence l, X, l∗ , Y, m gives us a path of length 4 between the vertices represented by l and m in GC . Finally, if A = B, we have two cases. If the plane determined by the lines l and m meets H∞ in a line tangent to C, then there is no line of LC meeting both l and m. Hence, using our earlier arguments, the distance between the corresponding vertices is 6. If however, the plane determined by the lines l and m meets H∞ in a line which is secant to C, then there is a third line l∗ ∈ LC meeting both l and m in two points, say X and Y . Hence, the sequence l, X, l∗ , Y, m represents a path of length 4 in GC . Therefore, the diameter of GC is either 4 or 6. When C is a hyperoval, there are no lines of H∞ that are tangent to C which implies the girth is 4. When C is an oval, we have tangent lines, and the girth is 6. ¥
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A construction using a polarity
We can construct a different example of a triangle-free line set using a symplectic polarity of P G(3, q), q odd. Recall that a symplectic polarity δ is induced by a skew-symmetric matrix. It is well-known that the points of P G(3, q) are all selfconjugate under δ; that is, they are incident with their polar image. The number of lines such that lδ = l is (q 2 + 1)(q + 1). Moreover, these lines have the property that there is a pencil of q + 1 lines through every point and in every plane of the space [5]. As a result these lines form a triangle-free set of lines of P G(3, q). Deleting a plane of P G(3, q) leaves us with a set of q 2 (q + 1) lines of AG(3, q) which are triangle-free. Let Lδ represent this set of lines of AG(3, q). We note that Lδ is unique up to projective equivalence [5]. As before, we let Mδ be the incidence matrix obtained from the points of AG(3, q) and lines of Lδ with the columns representing the points and rows representing the lines. Let C be the binary linear code with parity check matrix Mδ , and let C T be the code with parity check matrix MδT . Also, as before, we let Gδ be the bipartite graph with incidence matrix Mδ . Proposition 4.1 The graph Gδ has girth 8 and diameter 4. Proof. The girth is certainly at least 8 since a 6-cycle would contradict the trianglefree property. We again prove the statement by exhibiting an 8-cycle in the graph Gδ . For this, we fix the representation of the symplectic polarity. We use the skew-symmetric matrix 0 0 0 1 0 0 1 0 m= 0 -1 0 0 . -1 0 0 0
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Table 4: Codes from Symplectic Polarities in AG(3, q) q C CT
3 [27,6,12] [36,15,8]
5 [125,40,20] [150,65,18]
7 [343,126] [392,175]
9 [729,288] [810,369]
Table 5: Codes from Symplectic Polarities in P G(3, q) q C CT
3 [40,15,10] [40,15,8]
5 [156,65,20] [156,65,12]
7 [400,175] [400,175]
9 [820,369] [820,369]
Let (w, x, y, z) represent homogeneous coordinates for the points of P G(3, q). Then the lines defined by w = x = 0, w = y = 0, x = z = 0 and y = z = 0 are all fixed under the polarity induced by m. These four lines together with the points induced by (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) form our desired 8-cycle in the associated incidence graph. To show the diameter is 4, we take two arbitrary points P and Q of P G(3, q). If P and Q both lie on a line of Lδ , then the distance between P and Q is 2. If not, let π be any plane through P and Q. Then there exists a pencil of q + 1 lines of Lδ through a point, say R, of π. The point-line sequence P, P R, R, RQ, Q gives a path of length 4 between the vertices represented by P and Q. Now let l and m be two lines of Lδ . If l and m meet in a point, we are finished. If not, then there is a plane π through l meeting m in a point. The plane π contains q + 1 concurrent lines of Lδ through a point P . One of these lines, say l∗ meets m in a point Q. Hence, the sequence l, P, l∗ , Q, m gives us a path of length 4 between the vertices corresponding to l and m in Gδ . One can similarly find a path of length 3 between two vertices represented by a point P and line l. Hence, the diameter of Gδ is 4. ¥ Using Magma [2], we are able to determine the parameters of these codes when q is small. The results are summarized in Table 4. These codes appear to have the same length and dimension as the codes constructed from the ovals. However, the minimum distance appears to be larger for C T having jumped from 10 to 18 when q = 5, for example. We could also use the full set of points of P G(3, q) together with the lines of P G(3, q) fixed by δ to define our code. In this case, our length increases to q 3 + q 2 + q + 1. The codes obtained from these triangle-free line sets are described in Table 5. An optimal length 40 code has minimum weight 12, so we are only 2 away from being optimal [1]. We can make a conjecture as to the dimension of these codes based on the data in the table. Note that the information rate is roughly 1/2 if this conjecture is true. Conjecture 4.2 The codes above generated from a symplectic polarity in P G(3, q) are [q 3 + q 2 + q + 1, (q 3 + q)/2]-codes.
LDPC codes from triangle-free line sets
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Conclusion
The codes defined here provide nice geometric examples of LDPC codes with reasonably high girth. A natural question is whether or not these constructions can be generalized to higher dimensions. Certainly, the same construction using caps in P G(n, q) yields more codes of length q n . Unfortunately, the girth of the associated graphs remains fixed at 8. Triangles in projective spaces are particularly nice since they can only lie in a plane. Avoiding quadrilaterals (or, more generally, n-gons) seems more difficult since such geometric configurations are no longer necessarily planar. In order to generalize these constructions, one would have to avoid taking all the lines through the points of a special set in the hyperplane at infinity.
Acknowledgements The author would like to thank Aart Blokhuis for his very helpful comments on the proof of Theorem 2.5, and also Vera Pless for her helpful comments during the preparation of the paper.
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