Tuvi Etzion

Which codes have cycle-free Tanner graphs?

If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n/sup 2/). However, we show that cycle-free Tanner graphs cannot support good codes. Specifically, let C be an (n,k,d) linear code of rate R=k/n that can be represented by a Tanner graph without cycles. We prove that if R/spl ges/0.5 then d/spl les/2, while if R<0.5 then C is obtained from a code of rate /spl ges/0.5 and distance /spl les/2 by simply repeating certain symbols. In the latter case, we prove that d/spl les/[n/k+1]+[n+1/k+1]<2/R. Furthermore, we show by means of an explicit construction that this bound is tight for all values of n and k. We also prove that binary codes which have cycle-free Tanner graphs belong to the class of graph-theoretic codes, known as cut-set codes of a graph. Finally, we discuss the asymptotics for Tanner graphs with cycles, and present a number of open problems for future research.

Error-Correcting Codes in Projective Space

The projective space of order n over the finite field \BBFq, denoted here as Pq(n), is the set of all subspaces of the vector space \BBFqn . The projective space can be endowed with the distance function d(U, V) = dimU + dimV -2 dim(U ∩ V) which turns Pq(n) into a metric space. With this, an (n,M,d) code \BBC in projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d . Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n,M,d) code can correct t packet errors and ρ packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2ρ <; d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of “coding theory in projective space.” First, we present several new bounds on the size of codes in Pq(n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. We also present several specific constructions of codes and code families in Pq(n). Finally, we prove that nontrivial perfect codes in Pq(n) do not exist.

Error-Correcting Codes in Projective Spaces Via Rank-Metric Codes and Ferrers Diagrams

Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper, we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant-weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant-dimension code. The union of these codes is our final constant-dimension code. In particular, the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant-weight codes and the rank-metric codes. Finally, we use puncturing on our final constant-dimension codes to obtain large codes in the projective space which are not constant-dimension.

Constructions for optimal constant weight cyclically permutable codes and difference families

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, in. Eluding in optical code-division multiple-access communication systems and in constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback. In this paper we construct optimal constant weight cyclically permutable codes with length n, weight w, and a minimum Hamming distance 2w-2. Some of these codes coincide with the well-known design called a difference family. Some of the constructions use combinatorial structures with other applications in coding. >

Perfect binary codes: constructions, properties, and enumeration

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound the authors obtain a simple closed-form expression for the weight distribution of a perfect code. Furthermore, they prove that the characters of a perfect code satisfy certain constraints, and provide a sufficient condition for a binary code to be perfect. The latter result is employed to derive a generalization of the construction of Phelps (1983), which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions. Moreover, for any m/spl ges/4 the authors construct full-rank perfect binary codes of length 2/sup m/-1. These codes are obviously nonequivalent to any of the previously known perfect codes. Furthermore the latter construction exhibits the existence of full-rank perfect tilings. Finally, they construct a set of 2(2/sup cn/) nonequivalent perfect codes of length n, for sufficiently large n and a constant c=0.5-/spl epsiv/. Precise enumeration of the number of codes in this set provides a slight improvement over the results reported by Phelps. >

Constructions for perfect maps and pseudorandom arrays

A construction of perfect maps, i.e. periodic r*v binary arrays in which each n*m binary matrix appears exactly once, is given. A similar construction leads to arrays in which only the zero n*m matrix does not appear and to a construction in which only a few n*m binary matrices do not appear. A generalization to the nonbinary case is given. The constructions involve an interesting problem in shift-register theory. The solution is given for almost all the case of this problem. >

On Perfect Codes and Tilings: Problems and Solutions

Although nontrivial perfect binary codes exist only for length n = 2m -1 with

Codes and Anticodes in the Grassman Graph

m \ge 3

Algorithms for the generation of full-length shift- register sequences

and for length n=23, many interesting problems concerning these codes remain unsolved. Herein, we present solutions to some of these problems. In particular, we show that the smallest nonempty intersection of two perfect codes of length 2m -1 consists of two codewords, for all

Optimal constant weight codes over Z k and generalized designs

m \ge 3