Eirik Rosnes

An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices

In this work, we introduce an efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numerical examples. For instance, we have applied the algorithm to the well-known (3, 5)-regular (155, 64) Tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer. Also, we have verified that the minimum stopping set size of the (4896, 2474) Ramanujan-Margulis code is indeed 24, and that the corresponding multiplicity is exactly 204. Furthermore, we have applied the algorithm to the IEEE 802.16e LDPC codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes. Finally, as an application, we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity-check matrix in order to remove all stopping sets in the corresponding Tanner graph of size less than the minimum distance. An extensive case study of the (155, 64) Tanner code illustrates the usefulness of the algorithm, and we present a 110 times 155 redundant parity-check matrix for this code with no stopping sets of size less than the minimum distance. Simulation results of iterative decoding on the binary erasure channel show performance improvements for low-to-medium erasure probabilities when this redundant parity-check matrix is used for decoding.

Improved algorithms for the determination of turbo-code weight distributions

We discuss algorithms for determining exactly the lower terms of the weight distribution of a turbo code. Several improvements on the recently introduced algorithm by Garello et al. are outlined. The techniques presented in this letter improve the observed asymptotic complexity by a factor proportional to the information length. As an example, the improved algorithm is applied to the determination of the minimum distance of all universal mobile telecommunications system turbo codes. We further apply the improved algorithm to high-rate turbo codes using high-rate nonpunctured constituent codes. To reduce complexity, the constituent codes are represented by a minimal information bit-oriented trellis.

Optimum Distance Quadratic Permutation Polynomial-Based Interleavers for Turbo Codes

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Also, the recently proposed quadratic permutation polynomial (QPP) based interleavers by Sun and Takeshita (IEEE Trans. Inform. Theory, Jan. 2005) provide excellent performance for short-to-medium block lengths. In this work the minimum distance of turbo codes with QPP-based interleavers is considered in detail. Large tables of optimum (in terms of turbo code minimum distance and multiplicity) QPPs for turbo codes with 8-state and 16-state constituent codes are presented. The minimum distances are compared to existing results in the literature on dithered relative prime (DRP) interleavers. The optimality of the new tables makes them an excellent source of information to advance the understanding of permutation polynomial (PP) based interleavers

Turbo Decoding on the Binary Erasure Channel: Finite-Length Analysis and Turbo Stopping Sets

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size turbo stopping set which is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garello et al. in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.

Addendum to “An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices”

In an earlier transactions paper, Rosnes and Ytrehus presented an efficient algorithm for determining all stopping sets of low-density parity-check (LDPC) codes, up to a specified weight, and also gave results for a number of well-known codes including the family of IEEE 802.16e LDPC codes, commonly referred to as the WiMax codes. It is the purpose of this short paper to review the algorithm for determining the initial part of the stopping set weight spectrum (which includes the codeword weight spectrum), and to provide some improvements to the algorithm. As a consequence, complete stopping set weight spectra up to weight 32 (for selected IEEE 802.16e LDPC codes) can be provided, while in previous work only stopping set weights up to 28 are reported. In the published standard for the IEEE 802.16e codes there are two methods of construction presented, depending upon the code rate and the code length. We compare the stopping sets of the resulting codes and provide complete stopping set weight spectra (up to five terms) for all IEEE 802.16e LDPC codes using both construction methods.

On maximum length convolutional codes under a trellis complexity constraint

We look at convolutional codes with maximum possible code length for prescribed redundancy, conditioned on constraints on the free distance and on the bit-oriented trellis state complexity. Rate (n - 1)/n codes have been studied to some extent in the literature, but more general rates have not been studied much. In this work we consider convolutional codes of rate (n - r)/n, r ≥ 1. Explicit construction techniques for free distance dfree = 3 and 4 codes are described. For codes with r = 2, an efficient exhaustive search algorithm is outlined. For the more general case with r ≥ 2, a heuristic approach is suggested. Several new codes were found for r = 1 and in particular for r = 2 and 3. Compared to previously known codes of similar free distance and complexity constraints, the new codes have either strictly higher rate, or the same rate but smaller low distance multiplicities.

On the design of bit-interleaved turbo-coded modulation with low error floors

In this paper, we introduce an algorithm to optimize the performance in the error-floor region of bit-interleaved turbo-coded modulation (BITCM) on the additive white Gaussian noise channel. The key ingredient is an exact turbo code weight distribution algorithm producing a list of all codewords in the underlying turbo code of weight less than a given threshold. In BITCM, the information sequence is turbo-encoded, bit-interleaved, and mapped to signal points in a signal constellation. Using the union-bounding technique, we show that a well-designed bit interleaver is crucial to have a low error floor. Furthermore, the error-rate performance in the waterfall region depends on the bit interleaver, since the level of protection from channel noise on the bit level depends on the bit position and the neighboring bit values within the same symbol in the transmitted sequence. We observe a tradeoff between error-rate performance in the waterfall and error-floor regions, as illustrated by an extensive case study of a high-rate BITCM scheme. This tradeoff is typical in iterative decoding of turbo-like codes. The reported case study shows that it is possible to design bit interleavers with our proposed algorithm with equal or better performance in the waterfall region and superior performance in the error-floor region, compared with randomly generated bit interleavers. In particular, we were able to design BITCM schemes with maximum-likelihood decoding frame-error rates of 10-12 and 10 -17 at 2.6 and 3.8 dB away from unconstrained channel capacity, at spectral efficiencies of 3.10 and 6.20 b/s/Hz using square 16 and 256-quadrature amplitude modulation signal constellations, respectively

On the Minimum Distance of Turbo Codes With Quadratic Permutation Polynomial Interleavers

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementations. Also, the recently proposed quadratic permutation polynomial (QPP)-based interleavers by Sun and Takeshita have provided excellent performance for short-to-medium block lengths, and have been selected for the 3GPP LTE standard. In this paper, we derive some upper bounds on the best achievable minimum distance dmin of QPP-based conventional binary turbo codes (with tailbiting termination, or dual termination when the interleaver length N is sufficiently large) that are tight for larger block sizes. In particular, we show that the minimum distance is at most 2(2v+1 + 9), independent of the interleaver length, when there exists an inverse polynomial of degree two, where v is the degree of the primitive feedback and monic feedforward polynomials. However, allowing the QPP to have no inverse polynomials of degree two may give strictly larger minimum distances (and lower multiplicities). In particular, we provide several QPPs with no quadratic inverse for some of the 3GPP LTE interleaver lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly larger than 50. For instance, we have found a QPP for N = 6016 which gives an estimated dmin of 57. Furthermore, we provide the exact minimum distances and the corresponding multiplicities for all 3GPP LTE turbo codes (with dual termination) which shows that the best minimum distance is 51. Finally, we compute the best achievable minimum distances with QPP interleavers for all 3GPP LTE interleaver lengths N ≤ 4096, and compare these minimum distances with the ones we get when using the 3GPP LTE polynomials.

Frequency estimation of a single complex sinusoid using a generalized Kay estimator

This letter introduces a generalized version of Kays estimator for the frequency of a single complex sinusoid in complex additive white Gaussian noise. The Kay estimator is a maximum-likelihood (ML) estimator at high signal-to-noise ratio (SNR) based on differential phase measurements with a delay of one symbol interval. In this letter, the corresponding ML estimator with an arbitrary delay in the differential phase measurements is derived. The proposed estimator reduces the variance at low SNR, compared with Kays original estimator. For certain delay values, explicit expressions for the window function and the corresponding high SNR variance of the proposed generalized Kay (GK) estimator are presented. Furthermore, for some delay values, the window function is nearly uniform and the implementation complexity is reduced, compared with the original Kay estimator. For a delay value of two, we show that the variance at asymptotically high SNR approaches the Cramer-Rao bound as the sequence length tends to infinity. We also explore the effect of exchanging the order of summation and phase extraction for reduced-complexity reasons. The resulting generalized weighted linear predictor estimator and the GK estimator are compared with both autocorrelation-based and periodogram-based estimators in terms of computational complexity, estimation range, and performance at both low and high SNRs.

Improved algorithms for high rate turbo code weight distribution calculation

We discuss algorithms for determining exactly the lower terms of the weight distribution of a turbo code. Several improvements of the recently introduced algorithm by R. Garello et al. (see IEEE J. Select. Areas Commun., vol.19, p.800-12, 2001) are outlined. The techniques presented improve the observed asymptotic complexity by a factor proportional to the information length. We further apply the improved algorithm to high rate turbo codes using high rate non-punctured constituent codes. To reduce complexity, the constituent codes are represented by a minimal information bit oriented trellis.