Amin Shokrollahi

Representation theory for high-rate multiple-antenna code design

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels.

Capacity-achieving sequences for the erasure channel

This paper starts a systematic study of capacity-achieving (c.a.) sequences of low-density parity-check codes for the erasure channel. We introduce a class A of analytic functions and develop a procedure to obtain degree distributions for the codes. We show various properties of this class which help us to construct new distributions from old ones. We then study certain types of capacity-achieving sequences and introduce new measures for their optimality. For instance, it turns out that the right-regular sequence is c.a. in a much stronger sense than, e.g., the Tornado sequence. This also explains why numerical optimization techniques tend to favor graphs with only one degree of check nodes.

Design of provably good low-density parity check codes

We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. These codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. We further show that under suitable conditions the message densities fulfil a certain symmetry condition which we call the consistency condition and we present a stability condition which is the most powerful tool to date to bound/determine the threshold of a given family of low-density parity check codes.

Design of efficient erasure codes with differential evolution

The design of practical and powerful codes for protection against erasures can be reduced to optimizing solutions of a highly nonlinear constraint satisfaction problem. In this paper we will attack this problem using the differential evolution approach and significantly improve results previously obtained using classical optimization procedures.

Finite length analysis of LT codes

This paper provides an efficient method for analyzing the error probability of the belief propagation (BP) decoder applied to LT Codes. Each output symbol is generated independently by sampling from a distribution and adding the input symbols corresponding to the support of the sampled vector.

Raptor codes on symmetric channels

This paper extends the construction and analysis of Raptor codes originally designed in A. Shokrollahi (2004) for the erasure channel to general symmetric channels. We explicitly calculate the asymptotic fraction of output nodes of degree one and two for capacity-achieving Raptor codes, and discuss techniques to optimize the output degree distribution.

Using low density parity check codes in the McEliece cryptosystem

We examine the implications of using a low density parity check code (LDPCC) in place of the usual Goppa code in McElieces cryptosystem. Using a LDPCC allows for larger block lengths and the possibility of a combined error correction/encryption protocol.

Design of unitary space-time codes from representations of SU(2)

We explicitly construct an irreducible four-dimensional unitary representation of the Lie group SU(2) and use it for the construction of high-rate unitary space-time codes for four transmit antennas. Our construction calls for the design of such spherical codes in which the angle between any two points is well separated from 120 degrees. We give a partial solution to this restricted design problem and use numerical optimization techniques to optimize our solution.

Computing the performance of unitary space-time group codes from their character table

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment. Their success, however, depends on the design of cedes that achieve these promises. It is well known that unitary matrices can be used to design differentially modulated space-time codes. These codes have a particularly efficient description if they form a finite group under matrix multiplication. We show how to compute the parameters of such groups crucial for their use as space-time codes, using only the character table of the group. Since character tables for many groups are known and tabulated, this method could be used to quickly test, for a given group, which of its irreducible representations can be used to design good unitary space-time codes. We demonstrate our method by computing the eigenvalues of all the irreducible representations of the special linear group SL/sub 2/(F/sub q/) over a finite prime field F/sub q/ of odd characteristic, and study in detail the performance of a particular eight-dimensional representation of SL/sub 2/(F/sub 17/).

Verification decoding of raptor codes

In this paper we extend the double verification algorithm of Luby and Mitzenmacher to the class of Raptor codes, analyze it, and design Raptor codes that perform very well with respect to this algorithm