Upper Bounds on the Minimum Distance of Quasi-Cyclic LDPC codes Revisited
نویسندگان
چکیده
In this paper we investigate the minimum code distance of QC LDPC codes [1], [2], [3]. These codes form an important subclass of LDPC codes [4], [5]. These codes also are a subclass of protograph-based LDPC codes [6]. QC LDPC codes can be easily stored as their parity-check matrices can be easily described. Besides such codes have efficient encoding [7] and decoding [8] algorithms. All of these makes the codes very popular in practical applications. In [2] an upper bound on the minimum distance of QC LDPC codes is derived for the case when the base matrix has all the elements equal to one. In this case the minimum code distance is upper bounded by a quantity (m + 1)!, where m is a height of a base matrix and at the same time (due to the structure of the base matrix) the number of ones in a column of the base matrix. In [9] the results of [2] are generalized for the case of type-w QC LDPC codes (see Theorems 7 and 8 in [9]). Unfortunately these estimates can be applied only to a certain parity-check matrix. In this paper we obtain the upper bounds which are valid for any code from the ensemble of QC LDPC codes with the given degree distribution. This allows us to formulate the necessary condition for the minimum code distance of such codes to grow linearly with the code length. We consider only the case of so-called type-1 QC LDPC codes. Our contribution is as follows. Two upper bounds on the minimum distance of type-1 quasi-cyclic low-density paritycheck (QC LDPC) codes are derived. The necessary condition is given for the minimum code distance of such codes to grow linearly with the code length. The structure of the paper is as follows. In section II the preliminaries on QC LDPC codes are given. In section III the bounds are derived and analyzed.
منابع مشابه
A New Construction for LDPC Codes using Permutation Polynomials over Integer Rings
A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudorandom nature, and the new codes are quasi-cyclic. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 wer...
متن کاملMinimum Distances of the QC-LDPC Codes in IEEE 802 Communication Standards
This work applies earlier results on Quasi-Cyclic (QC) LDPC codes to the codes specified in six separate IEEE 802 standards, specifying wireless communications from 54 MHz to 60 GHz. First, we examine the weight matrices specified to upper bound the codes’ minimum distance independent of block length. Next, we search for the minimum distance achieved for the parity check matrices selected at ea...
متن کاملFrom Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes
Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density paritycheck (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we c...
متن کاملCyclic and Quasi-Cyclic LDPC Codes on Row and Column Constrained Parity-Check Matrices and Their Trapping Sets
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descenda...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1401.2120 شماره
صفحات -
تاریخ انتشار 2014