A. R. Calderbank

Baseband trellis codes with a spectral null at zero

A method is described for modifying classical N-dimensional trellis codes to provide baseband codes that combine a spectral null at DC with significant coding gain. The information rate of the classical code is decreased by one bit, and this extra redundancy is used to keep the running digital sum bounded. Equivalently, if the rate is held constant, then twice as many signal points are needed, causing a power penalty of 6/N dB. Baseband trellis codes are presented for several information rates together with complete spectral plots and performance comparisons. A method of constructing baseband codes with multiple spectral nulls is also described. >

A 2-adic approach to the analysis of cyclic codes

This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over Z/sub 2(a)/, a/spl ges/2, the ring of integers modulo 2/sup a/. It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over Z/sub 2(a)/ that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2/sup a/ appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over Z/sub 2(a)/ are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z/sub 4/ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsart-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over Z/sub 4/ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48.

Synchronizable codes for the optical OPPM channel

Random overlapping pulse-position modulation (OPPM) sequences result in an unrecoverable error floor on both the probability of erroneous synchronization and the probability of symbol error when only chip synchronization is present. It is known, however, that for a given sequence length M, a subset of the set of all possible sequences is synchronizable in the sense that in the absence of noise, the receiver can correctly symbol synchronize by observing M or more symbol intervals. The authors design finite-state machines and codes over a J-ary alphabet, which produce sequences with the property that every subsequence of length L is synchronizable. Some of the codes, in addition to being synchronizable, produce a coding gain. For an optical Poisson channel the authors introduce joint synchronization and detection algorithms that utilize the memory in the encoded sequences to produce joint estimates of timing and sequences. Their performance is analyzed through simulations and analytical results. >

On a conjecture of Helleseth regarding pairs of binary m-sequences

Binary m-sequences are maximal-length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communication. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. This correspondence addresses a conjecture made by Helleseth in 1976, that if m is a power of 2, then there are no pairs of binary m-sequences with a 3-valued crosscorrelation function. This conjecture is proved under the assumption that the three correlation values are symmetric about -1.

Inequalities for covering codes

Any code C with covering radius R must satisfy a set of linear inequalities that involve the Lloyd polynomial L/sub R/(x); these generalize the sphere bound. Syndrome graphs associated with a linear code C are introduced to help keep track of low-weight vectors in the same coset of C (if there are too many such vectors C cannot exist). Illustrations show that t(17, 10)=3 and t(23, 15)=3 where t(n, k) is the smallest covering radius of any (n, k) code. >

A multilevel approach to the design of DC-free line codes

A multilevel approach to the design of DC-free line codes is presented. The different levels can be used for different purposes, for example, to control the maximum accumulated charge or to guarantee a certain minimum distance. The advantages of codes designed by this method over similar codes are the improved run-length/accumulated-charge parameters, higher transmission rate, and the systematic nature of the code construction. The multilevel structure allows the redundancy in the signal selection procedure to be allocated efficiently among the different levels. It also allows the use of suboptimal staged decoding procedures that have performance/complexity advantages over maximum-likelihood decoding. >

A strengthening of the Assmus-Mattson theorem

Let w/sub 1/=d,w/sub 2/,...,w/sub s/ be the weights of the nonzero codewords in a binary linear (n,k,d) code C, and let w/sub 1/, w/sub 2/, ..., w/sub 3/, be the nonzero weights in the dual code C1. Let t be an integer in the range 0 or=d+4 then either the words of any nonzero weight w/sub i/ form a (t+1)-design or else the codewords of minimal weight d form a (1,2,...,t,t+2)-design. If in addition C is self-dual with all weights divisible by 4 then the codewords of any given weight w/sub i/ form either a (t +1)-design or a (1,2,...,t,t+2)-design. The proof avoids the use of modular forms. >

Extending the T-design Concept

Let B be a family of k-subsets of a v-set V , with 1 less-than-or-equal-to k less-than-or-equal-to v/2. Given only the inner distribution of B i.e., the number of pairs of blocks that meet in j points (with j = 0, 1, ... , k), we are able to completely describe the regularity with which B meets an arbitrary t-subset of V , for each order t (with 1 less-than-or-equal-to t less-than-or-equal-to v/2). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters v, k, t. The main regularity parameter is the dimension of a well-defined subspace of R(t+1), called the t-form space of B. (This subspace coincides with R(t+1) if and only if B is a t-design.) We show that the t-form space has the structure of an ideal, and we explain how to compute its canonical generator.

On error-correcting codes and invariant linear forms

Given a code C, invariant linear forms are used to study the designs afforded by codewords of a fixed weight. The most important theorem relating codes and designs is due to Assmus and Mattson [J. Combin. Theory, 6 (1969), pp. 122–151], and this theorem is extended in different ways. For extremal self dual codes over the fields

Channel coding for cochannel interference suppression in wireless communication systems

mathbb{F}_2