Minimality for puncturing convolutional codes

This paper investigates encoders optimization for the Hamming weight after periodic puncturing, and discusses minimality issues that may affect the performance of the punctured encoders. Periodically puncturing a minimal encoder produces a higher rate encoder that may or may not be minimal. If it is not minimal, it may have a zero-output loop and it may be catastrophic. A code search can use a fast algorithm to determine whether an encoder's state diagram has a zero-output loop under periodic symbol puncturing, and a proposed method to assess the performance of codes with a zero-output loop that are not catastrophic. As an example, the paper optimizes rate-1/4 unpunctured codes for Hamming weight under both bit-wise and symbol-wise periodic puncturing. Code tables and simulation results are included

Generalized punctured convolutional codes

Abstract-This letter introduces the class of generalized punctured convolutional codes (GPCCs), which is broader than and encompasses the class of the standard punctured convolutional codes (PCCs). A code in this class can be represented by a trellis module, the GPCC trellis module, whose topology resembles that of the minimal trellis module. The GPCC trellis module for a PCC is isomorphic to the minimal trellis module. A list containing GPCCs with better distance spectrum than the best known PCCs with same code rate and trellis complexity is presented

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

A code trellis is a graphical representation of a code, block or convolutional, in which every path represents a codeword (or a code sequence for a convolutional code). This representation makes it possible to implement Maximum Likelihood Decoding (MLD) of a code with reduced decoding complexity. The most well known trellis-based MLD algorithm is the Viterbi algorithm. The trellis representation was first introduced and used for convolutional codes [23]. This representation, together with the Viterbi decoding algorithm, has resulted in a wide range of applications of convolutional codes for error control in digital communications over the last two decades. There are two major reasons for this inactive period of research in this area. First, most coding theorists at that time believed that block codes did not have simple trellis structure like convolutional codes and maximum likelihood decoding of linear block codes using the Viterbi algorithm was practically impossible, except for very short block codes. Second, since almost all of the linear block codes are constructed algebraically or based on finite geometries, it was the belief of many coding theorists that algebraic decoding was the only way to decode these codes. These two reasons seriously hindered the development of efficient soft-decision decoding methods for linear block codes and their applications to error control in digital communications. This led to a general belief that block codes are inferior to convolutional codes and hence, that they were not useful. Chapter 2 gives a brief review of linear block codes. The goal is to provide the essential background material for the development of trellis structure and trellis-based decoding algorithms for linear block codes in the later chapters. Chapters 3 through 6 present the fundamental concepts, finite-state machine model, state space formulation, basic structural properties, state labeling, construction procedures, complexity, minimality, and sectionalization of trellises. Chapter 7 discusses trellis decomposition and subtrellises for low-weight codewords. Chapter 8 first presents well known methods for constructing long powerful codes from short component codes or component codes of smaller dimensions, and then provides methods for constructing their trellises which include Shannon and Cartesian product techniques. Chapter 9 deals with convolutional codes, puncturing, zero-tail termination and tail-biting.Chapters 10 through 13 present various trellis-based decoding algorithms, old and new. Chapter 10 first discusses the application of the well known Viterbi decoding algorithm to linear block codes, optimum sectionalization of a code trellis to minimize computation complexity, and design issues for IC (integrated circuit) implementation of a Viterbi decoder. Then it presents a new decoding algorithm for convolutional codes, named Differential Trellis Decoding (DTD) algorithm. Chapter 12 presents a suboptimum reliability-based iterative decoding algorithm with a low-weight trellis search for the most likely codeword. This decoding algorithm provides a good trade-off between error performance and decoding complexity. All the decoding algorithms presented in Chapters 10 through 12 are devised to minimize word error probability. Chapter 13 presents decoding algorithms that minimize bit error probability and provide the corresponding soft (reliability) information at the output of the decoder. Decoding algorithms presented are the MAP (maximum a posteriori probability) decoding algorithm and the Soft-Output Viterbi Algorithm (SOVA) algorithm. Finally, the minimization of bit error probability in trellis-based MLD is discussed

Periodic state-space representations of periodic convolutional codes

In this paper we study the representation of periodically time-varying convolutional codes by means of periodic input-state-output models. In particular, we focus on period two and investigate under which conditions a given two-periodic convolutional code (obtained by alternating two time-invariant encoders) can be represented by a periodic input-state-output system. We first show that one cannot expect, in general, to obtain a periodic input-state-output representation of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic state-space realization that is minimal. Finally, examples to illustrate the results are presented.publishe

Turbo Codes with Non-Uniform Constellation

This paper presents parallel concatenated turbo codes that employ a non-uniform constellation to achieve shaping gain. The output signal approximates the Gaussian distribution by using equally likely signals with unequal spacing (a non-uniform constellation). The small distance of points near the center of the constellation may lead to a small overall free distance and thus a high error floor for turbo codes. We avoid this situation by a two-step design procedure, that first creates an interleaver, and then identifies the constituent encoders that maximize the turbo code free distance. Simulation results for 4 bits/sec/Hz show that this use of shaping can offer an improvement of approximately 0.2 dB for turbo codes

Asymptotically good convolutional codes with feedback encoders

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1997.Includes bibliographical references (leaves 47-49).by Peter J. Sallaway.M.Eng

The Dynamics of Group Codes: Dual Abelian Group Codes and Systems

Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act as the character groups of the state spaces of C^\perp. The controllability properties of C are the observability properties of C^\perp. In particular, C is (strongly) controllable if and only if C^\perp is (strongly) observable, and the controller memory of C is the observer memory of C^\perp. The controller granules of C act as the character groups of the observer granules of C^\perp. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an "end-around" controller granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200