We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NP-hard, and the corresponding decision problem is NP-complete. This constitutes a proof of the conjecture Bedekamp, McEliece, van Tilborg, dating back to 1978. Extensions and applications of this result to other problems in coding theory are discussed.

A complete classification of the perfect binary oneerror-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. Östergård and O. Pottonen, “The perfect binary one-error-correcting codes of length 15: Part I—Classification, ” submitted for publication]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via i-components, as it turns out that all but two full-rank codes can be obtained through a series of transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.

Basic structural properties of tail-biting trellises are investigated. We start with rigorous definitions of various types of minimality for tail-biting trellises.

We present an efficient algorithm for computing the minimal trellis for a group code over a finite Abelian group, given a generator matrix for the code. We also show how to cornpure a succinct representation of the minimal trellis for such a code, andpresent algorithms that use this information to efficiently compute local descriptions of the minimal trellis. This extends the work of Kschischang and Sorokine, who handled the case of linear codes over fields. An important application of our algorithms is to the construction qf minireal trellises for lattices. A key step in our work is handling codes over cyclic groups C'p, where p is a prime. Such a code can be viewed as a submodule over the ring Zp. Because of the presence of zero-divisors in the ring, submodules do not share the useful properties of vector spaces. We get around this difficulty by restricting the notion of linear combination to p-linear combination, and introducing the notion of a p-generator equence, which enjoys properties similar to that of a generector matrix for a vector space.

Abstract — A number of authors have recently considered iterative soft-in soft-out (SISO) decoding algorithms for classical linear block codes that utilize redundant Tanner graphs. Jiang and Narayanan presented a practically realizable algorithm that applies only to cyclic codes while Kothiyal et al. presented an algorithm that, while applicable to arbitrary linear block codes, does not imply a low-complexity implementation. This work first presents the aforementioned algorithms in a common framework and then presents a related algorithm- random redundant iter-ative decoding- that is both practically realizable and applicable to arbitrary linear block codes. Simulation results illustrate the successful application of the random redundant iterative decoding algorithm to the extended binary Golay code. Addi-tionally, the proposed algorithm is shown to outperform Jiang and Narayanan’s algorithm for a number of Bose-Chaudhuri-Hocquenghem (BCH) codes. I.

Abstract not found

Abstract not found

Two broad classes of graphical modeling problems for codes can be identified in the literature: constructive and extractive problems. The former class of problems concern the construction of a graphical model in order to define a new code. The latter class of problems concern the extraction of a graphical model for a (fixed) given code. The design of a new low-density parity-check code for some given criteria (e.g. target block length and code rate) is an example of a constructive problem. The determination of a graphical model for a classical linear block code which implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem. This work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes. The primary focus of this work is a study of the space of all graphical models for a (fixed) given code. The tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound: the tree-inducing cut-set bound. The proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools (e.g. the Cut-Set Bound) and can be viewed as a generalization of the square-root bound for tail-biting trellises to graphical models with arbitrary cyclic topologies. Searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations which are shown to span this space. Finally, heuristics for extracting novel graphical models for linear block codes using these transformations are investigated.

We consider various computational techniques in algebraic coding theory along two lines of work. First we investigate optimization of non-linear codes by relaxing minimum distance constraints, developing, in the process, two algorithms for improving a given non-linear code and a method of visualizing algebraic codes in three dimensions. Secondly, we study the Generalized Lexicographic Construction, and show that it produces as special cases the lexicodes and derivatives with properties such as trellis-orientation, trellis-state boundedness, and local optimality. We implement algorithms for generating these families of codes and, in the process, improve upon work by Conway and Sloane, Brualdi and Pless, Kschischang and Horn, and Zhang.

In this correspondence, we give a new definition of generalized Hamming weights of nonlinear codes and a new interpretation connected with it. These generalized weights are determined by the entropy/length profile of the code. We show that this definition characterizes the performance of nonlinear codes on the wire-tap channel of type II. The new definition is invariant under translates of the code, it satisfies the property of strict monotonicity and the generalized Singleton bound. We check the relations between the generalized weight hierarchies of Rlinear codes and their binary image under the Gray map. We also show that the binary image of a R-linear code is a symmetric, not necessarily rectangular code. Moreover, if this binary image is a linear code then it admits a twisted squaring construction.