Weights of a class of projective geometry codes

2 - Combinatorial Constructions and Coding

Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

In this paper we propose an analytical method to evaluate the performance of one step majority logic decoders constructed from faulty gates. We analyze the decoder under the assumption that the gates fail independently. We calculate the average bit error probability of such a decoder and apply the method to the special case of projective geometry codes. The method, however, applies to any regular low-density parity-check code of girth at least six but the calculations are much simpler for the projective geometry codes. We present results for the bit error rate performance of four codes from projective planes.

The class of polynomial codes introduced by Kasami et al. has considerable inherent algebraic and geometric structure. It has been shown that this class of codes and their dual codes contain many important classes of cyclic codes as subclasses, such as BCH codes, Reed-Solomon codes, generalized Reed-Muller codes, projective geometry codes, and Euclidean geometry codes. The purpose of this paper is to investigate further properties of polynomial codes and their duals. First, majority-logic decoding for the duals of certain primitive polynomial codes is considered. Two methods of forming nonorthogonal parity-check sums are presented. Second, the maximality of Euclidean geometry codes is proved. The roots of the generator polynomial of an Euclidean geometry code are specified.

The design of superimposed codes for the multiaccess OR-channel is considered. The performance of constant-weight codes when used as superimposed codes is investigated. Several constructions for constant-weight codes are compared: affine geometry codes, projective geometry codes, and codes obtained by code concatenation. A comparison to the sphere packing bound and the Johnson bound is made. >

Polynomial codes and their dual codes as introduced by Kasami, Lin, and Peterson have considerable algebraic and geometric structure. It has been shown that these codes contain many well-known classes of cyclic codes as subclasses, such as BCH codes, projective geometry codes (PG codes), Euclidean geometry codes (EG codes), and generalized Reed-Muller codes (GRM codes). In this paper, combinatorial expressions for the number of information symbols and parity-check symbols in polynomial codes are derived. The results are applied to two important subclasses of codes, the PG codes and EG codes.

Projective Geometry and linear codes

In this paper we focus our attention on a family of finite geometry codes, called type-I projective geometry low-density parity-check (PG-LDPC) codes, that are constructed based on the projective planes PG{2,q). In particular, we study their minimal codewords and pseudo-codewords, as it is known that these vectors characterize completely the code performance under maximum-likelihood decoding and linear programming decoding, respectively. The main results of this paper consist of upper and lower bounds on the pseudo-weight of the minimal pseudo-codewords of type-I PG-LDPC codes.

In this paper, an improved decoding algorithm for codes that are constructed from finite geometries is introduced. The application of this decoding algorithm to Euclidean geometry (EG) and projective geometry (PG) codes is further discussed. It is shown that these codes can be orthogonalized in less than or equal to three steps. Thus, these codes are majority-logic decodable in no more than three steps. Our results greatly reduce the decoding complexity of EG and PG codes in most cases. They should make these codes very attractive for practical use in error-control systems.

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We introduce a class of finite systems models of gene regulatory networks exhibiting behavior of the cell cycle. The model is an extension of a Boolean network model. The system spontaneously cycles through a finite set of internal states, tracking the increase of an external factor such as cell mass, and also exhibits checkpoints in which errors in gene expression levels due to cellular noise are automatically corrected. We present a 7-gene network based on Projective Geometry codes, which can correct, at every given time, one gene expression error. The topology of a network is highly symmetric and requires using only simple Boolean functions that can be synthesized using genes of various organisms. The attractor structure of the Boolean network contains a single cycle attractor. It is the smallest nontrivial network with such high robustness. The methodology allows construction of artificial cell cycle gene regulatory networks with the number of phases larger than in natural cell cycle.

In the last decade two old methods for decoding linear block codes have gained considerable interest, iterative decoding as first described by Gallager in [1] and list decoding as introduced by Elias [2]. In particular iterative decoding of low-density parity-check (LDPC) codes, has been an important subject of research, see e.g. [3] and the references therein. "Good" LDPC codes are often randomly generated by computer, but recently codes with an algebraic or geometric structure have also been considered e.g [3] and [4]. The performance of the iterative decoder is typically studied by simulations and a theoretical analysis is more difficult. In this paper we combine the two decoding methods and present an iterative list decoding algorithm. In particular we apply this decoder to a class of LDPC codes from finite geometries and show that the (73, 45, 10) projective geometry code can be maximum likelihood decoded with low complexity. Moreover the list decoding approach enables us to give a complete analysis of the performance in this case. We also discuss the performance of the list bit-flipping algorithm for longer LDPC codes. We consider hard-decision iterative decoding of a binary (n, k, d) code. For a received vector, y, we calculate an extended syndrome s = Hy′, where H is a parity check matrix, but usually has more than n - k rows. Let r denote the length of the syndrome. The idea of using extended syndromes was also used in [5]. Our approach is based on one of the common versions of bit flipping (BF) [3], where the schedule is such that the syndrome is updated after each flip. In each step we flip a symbol chosen among those positions that reduce the weight of the extended syndrome, which we refer to briefly as the syndrome weight, u. A decoded word is reached when u = 0. In this paper we consider a variation of the common algorithm in the form of a tree-structured search. Whenever there is a choice between several bits, all possibilities are tried in succession. The result of the decoding algorithm is, in general, a list of codewords, obtained as leaves of the search tree. This form of the bit flipping algorithm leads naturally to a solution in the form of a list of codewords at the same smallest distance from y [6]. This list decoding concept is somewhat different from list decoding in the usual sense of all codewords within a certain distance from y. The paper is a continuation of [7] including results on long codes from [8].

In this paper, we propose a family of quantum synchronizable codes from repeated-root cyclic codes and constacyclic codes. This family of quantum synchronizable codes are based on (λ(u + v)|u - v) construction which is constructed from constacyclic codes. Under this construction, we enrich the varieties of valid quantum synchronizable codes. We also prove that the obtained quantum synchronizable codes can achieve maximum synchronization error tolerance. Furthermore, quantum synchronizable codes based on (λ(u + v)|u - v) construction are shown to be able to have a better capability in correcting bit errors than those from projective geometry codes.

The maximum likelihood decoding problem for linear binary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k)</tex> codes is reformulated as a continuous optimization problem in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -dimensional solid cube. We obtain a near optimum solution of this problem by use of a simple gradient local optimization algorithm. Computer simulation results are presented for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(21,11)</tex> projective geometry code and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(47,23)</tex> quadratic-residue code.

We present an iterative list decoding algorithm for low-density parity-check (LDPC) codes. In particular we apply this decoder to a class of LDPC codes from finite geometries and show that the (73,45,10) projective geometry code can be maximum-likelihood (ML) decoded with low complexity. Moreover, the list decoding approach enables us to give a theoretical analysis of the performance. We also consider list bit-flipping (BF) decoding of longer LDPC codes.