## LDPC codes from triangle-free line sets (2004)

Venue: | Designs, Codes, and Cryptog |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Mellinger04ldpccodes,

author = {Keith E. Mellinger and Keith E. Mellinger},

title = {LDPC codes from triangle-free line sets},

journal = {Designs, Codes, and Cryptog},

year = {2004},

volume = {32},

pages = {341--350}

}

### OpenURL

### 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. 1

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Citation Context ...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 representati... |

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Citation Context ... 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 whi... |

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Citation Context ...alled 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... |

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Citation Context ...polar image. The number of lines such that l δ = 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 ... |

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Citation Context ...ctor v is orthogonal to every row of the parity check matrix M T C . 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... |

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Citation Context ...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, x 2 ) : x ∈ GF (q)} ∪ {(0, 0, 0, 1)}. �s4 Mellinger Let MC be the associated q 2 (q + 1) × q 3 incidence matrix over t... |

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Citation Context ... 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], with... |

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Citation Context ...mentation 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 1s2 Mellinger 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. 2 Bounds on triangle-free line sets We start by loo... |

13 |
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Citation Context ...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... |