Upper bounds on the minimum distance for turbo codes using CPP interleavers

Recently, it was shown that the dithered relative prime interleavers and quadratic permutation polynomial (QPP) interleavers can be expressed in terms of almost regular permutation (ARP) interleavers. In this paper, the conditions for a QPP interleaver to be equivalent to an ARP interleaver are extended for cubic permutation polynomial (CPP) interleavers. It is shown that the CPP interleavers are always equivalent to an ARP interleaver with disorder degree greater than one and smaller than the interleaver length, when the prime factorization of the interleaver length contains at least one prime number to a power higher than one and it fulfills the conditions for which there are true CPPs for the considered length. When the prime factorization of the interleaver length contains only prime numbers to the power of one, with at least two prime numbers $p_{i}$ , fulfilling the conditions $p_{i}>3$ and $3 \nmid (p_{i}-1)$ , values of disorder degree smaller than the interleaver length are possible under some conditions on the coefficients of the second and third degree terms of the CPP.

In this chapter we present upper bounds on the minimum distance of turbo codes using QPP interleavers obtained by Eirik Rosnes in (2012), as well as an upper bound on the minimum distance of turbo codes using PP-based interleaver of any degree given in Ryu et al. (2015). We note that most of these upper bounds are partial, meaning that they are valid only for certain interleaver lengths complying with certain constraints on their prime decomposition. A single upper bound is general (see Theorem 5.10, Eq. (5.112)), meaning that it holds for any interleaver length, but there is a constraint on the minimum degree of the inverse polynomials to be equal to two (i.e. the QPP interleaver must have an inverse QPP (Ryu and Takeshita 2006)). Because some of the above mentioned upper bounds use inverse polynomials of QPPs, in the first section of this chapter we present the necessary and sufficient condition for a QPP to admit at least one quadratic inverse, derived by Jonghoon Ryu and Oscar Y. Takeshita in Ryu and Takeshita (2006). At the end of the first section we give, without proof, a theorem from Ryu (2007), Ryu and Takeshita (2011) which allows us to determine all inverse permutation polynomials of a QPP of the smallest degree.

Turbo codes are composed of two or more convolutional codes. Therefore, in Sect. 2.1 we present the manner in which they can be described (i.e. generator matrix, state diagram and trellis diagram). Notions such as minimum Hamming distance, weight of a bit sequence, weight enumeration functions are assumed to be known (Proakis 1995; Trifina and Munteanu 2008). Then, in Sect. 2.2, we give the structure of a turbo code with two component convolutional codes. In Sect. 2.3 we present five methods to terminate the trellises for the two component convolutional codes of a turbo code. This operation is required to obtain a good error rate performance. Finally, in Sect. 2.4, we describe Garello’s algorithm (Garello et al. 2001) for calculating the distance spectrum of a turbo code with a particular interleaver.

Runlength-limited (RLL) and digital-sum-limited (DSL) codes are considered. For these codes the finite and asymptotic lower bounds on achievable rates for the given minimum Hamming distance are derived. Using generating functions and trellis diagram techniques, the authors prove the existence of RLL- and DSL-codes of rate R and minimum distance d= delta n, such that R>or=2.log lambda /sub 1/(0)-min(log lambda /sub 1/(z)- delta .log z), 0<or=z<or=1, for large enough code length n. The value lambda /sub 1/(z) is the maximum modulus eigenvalue of a certain matrix that is dependent on the limitations and some parameter z. The bounds for the limited codes improve the lower bound of H. C. Ferreira (1984).<<ETX>>

The encoding, decoding and performance analysis of concatenated partial unit memory (PUM) code structures for capacity approaching performance are investigated. PUM codes are known for their excellent distance properties and lower decoding complexity compared to equivalent multi-memory convolutional codes. Two capacity approaching concatenated structures are considered: turbo codes (TCs) and woven turbo codes (WTCs), both initially proposed with component convolutional codes. TCs consist of a number of parallel concatenated encoders. WTCs were originally proposed with outer warp, that is a number of outer encoders are parallel concatenated to one inner encoder. WTCs are also constructed with inner warp, i.e. one outer encoder is parallel concatenated to a number of inner encoders. An iterative max-log-maximum a posteriori decoding scheme is proposed for decoding PUM codes having multiple-input parallel branches in the trellis. The distance properties and bit error rate performance of the novel concatenated PUM codes are compared with known concatenated convolutional codes. It is observed that PUM structures, both TC and WTC with inner and outer warp, have better distance properties and hence improved performance over their equivalent convolutional counterparts. Performance as close as 1.25 dB from the Shannon limit is obtained for concatenated PUM structures with an interleaver size of only 4800 bits.

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.

This paper proposes a novel alternative to the Logarithmic Bahl-Cocke-Jelinek-Raviv (Log-BCJR) algorithm for turbo decoding, yielding significantly improved processing throughput and latency. While the Log-BCJR processes turbo-encoded bits in a serial forwards-backwards manner, the proposed algorithm operates in a fully-parallel manner, processing all bits in both components of the turbo code at the same time. The proposed algorithm is compatible with all turbo codes, including those of the LTE and WiMAX standards. These standardized codes employ odd-even interleavers, facilitating a novel technique for reducing the complexity of the proposed algorithm by 50%. More specifically, odd-even interleavers allow the proposed algorithm to alternate between processing the odd-indexed bits of the first component code at the same time as the even-indexed bits of the second component, and vice-versa. Furthermore, the proposed fully-parallel algorithm is shown to converge to the same error correction performance as the state-of-the-art turbo decoding algorithm. Owing to its significantly increased parallelism, the proposed algorithm facilitates throughputs and latencies that are up to 6.86 times superior to those of the state-of-the art algorithm, when employed for the LTE and WiMAX turbo codes. However, this is achieved at the cost of a moderately increased computational complexity and resource requirement.

A reversible data hiding scheme provides security for a secret message by embedding it in a cover image. After extracting the secret message at the receiver, the scheme restores the original message and the cover image without any distortion. However, data transmitted by the reversible data hiding scheme is vulnerable to the noise existing on the communications channel. A reversible data hiding scheme using turbo code is proposed in this paper to address that problem. The proposed reversible data hiding integrates modified matrix embedding, turbo code, and syndrome embedding to provide improved embedding capacity, security, and error correction. Modified matrix embedding hides the secret message in the cover image. The resulting stego-image is then encrypted and encoded by the turbo code. Turbo code deploys a secret key interleaver that shuffles the incoming bits using elliptic curve arithmetic and a secret key. The exact recovery of the cover image and the secret message by the receiver depends on the user-specific secret key. Modified matrix embedding hides a large number of secret message bits, which distorts the stegoimage. However, encrypting the stego-image with the secret key interleaver makes the embedding invisible. The proposed syndrome embedding transmits the locations of the modified bits in the cover image, and enables the receiver to reverse the effects of embedding. Simulation results with the proposed scheme demonstrate a high embedding capacity and improved error correction performance of 10-4 for an additive white Gaussian noise (AWGN) channel at an SNR of 2dB. It also successfully recovers the exact cover image and the secret message, compared to existing schemes such as matrix embedding and reversible data hiding in the encrypted domain. Moreover, the results depict high resistance to brute force attacks, statistical attacks, noise attacks, and cropping attacks.

The fifth-generation (5G) wireless communication system promises to revolutionize global connectivity by offering ultra-high-speed data transfer, massive device connectivity, and low latency. However, achieving these objectives demands efficient spectrum allocation and error-correcting mechanisms. The scarcity of radio spectrum and interference management are key challenges, necessitating optimized solutions for spectrum allocation. Additionally, reliable communication over noisy channels requires robust error-correcting codes. Optimized turbo codes, known for their iterative decoding capabilities, have emerged as a viable solution to enhance error resilience and throughput in 5G networks. This research proposes an integrated approach to optimize spectrum allocation and turbo code performance. The spectrum allocation employs a dynamic multi-objective optimization model based on machine learning algorithms, prioritizing fairness, quality of service (QoS), and interference minimization. Simultaneously, an improved turbo coding algorithm utilizing adaptive puncturing and interleaving strategies enhances data integrity. Simulations conducted in a heterogeneous 5G environment demonstrate significant performance improvements. The optimized turbo codes achieve a Bit Error Rate (BER) of 10-5 at a Signal-to-Noise Ratio (SNR) of 2.5 dB, outperforming conventional turbo codes by 40%. The proposed spectrum allocation strategy enhances spectral efficiency by 25%, ensuring equitable resource distribution and improved QoS. This integrated framework highlights the potential for scalable and efficient 5G systems, addressing the dual challenges of spectrum scarcity and error correction.

A new type of interleaver, block-random interleaver, is proposed and compared with the principal types of interleavers on the basis of the BER (bit error rate) performances of the turbo code. A recursive systematic convolutional code, 1/3 rate, unpunctured turbo code was used. The convolutional code has the constraint length, K=4 with generator matrix [1, 15/13], in octal representation. The MAP algorithm, 12 iterations were used. A log likelihood ratio (LLR) stop criterion was selected. The BER and FER (frame error rate) for different interleaver lengths were obtained.

We propose and compare some design criteria for the search of good rate-compatible punctured turbo (RCPT) code families. The considerations presented by S. Benedetto et al. (see IEEE Trans. on Commun., vol.46, no.9, p.1101-5, 1998) to find "best" component encoders for turbo code construction are extended to find the "best" rate-compatible puncturing patterns for a given interleaver length k. This approach is shown to be the optimal one, both in the maximum-likelihood sense (using transfer function bounds) and in the iterative decoding sense (through simulation results). A rate-compatibility restriction to the puncturing rule is used, implying that all the code bits of a high-rate punctured code are used by the lower rate codes. The two main applications of this technique are its use in modified type II ARQ/FEC schemes and to achieve unequal error protection of an information sequence.

This paper is aimed at designing of an effective interleaver for parallel concatenated coding schemes which utilizes the weight distribution as design criterion. We present a method here for designing an interleaver for achieving a minimum effective free distance of the code word so that error floor can be decreased. The method is expected to achieve a separation of radicN in the output code word. The proposed design has the advantage that the complexity grows linearly with the interleaver length.

A full-diversity full-rate (FDFR) design was developed recently, to enable an uncoded layered space-time (LST) system to achieve full-diversity (N/sub t/N/sub r/) and full-rate (N/sub t/ symbols per channel use) simultaneously, for any number of transmit antennas N/sub t/ and receive antennas N/sub r/. We investigate the performance of a coded FDFR system obtained by concatenating an error control coding (ECC) module and an FDFR module with a random interleaver in between. Iterative decoding is performed at the receiver. With R/sub c/ denoting the ECC rate, and d the minimum Hamming distance, an over-all transfer rate of R/sub c/N/sub t/ symbols per channel use and a full diversity order dN/sub t/N/sub r/ are achieved. Compared with coded V-BLAST, without sacrificing rate the coded FDFR system offers evident performance improvement when relatively weak codes are used. As N/sub r/ increases, even such a strong code as rate 1/2 turbo codes can benefit from FDFR. Specifically, 1.5 dB gain over coded V-BLAST is obtained in a 2 /spl times/ 2 antenna setup when convolutional codes or rate 3/4 turbo codes are used. 0.5 dB gain is offered in a 2 /spl times/ 5 setup when rate 1/2 turbo code is used. The price paid is increased complexity.

In this letter, we obtain lower and upper bounds on the minimum distance $d_{\min }$ of low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value $a_{\max }$ , exhaustively. We also derive a lower bound $L_{ne}$ on the weight of non-elementary code words. By performing the search with $a_{\max } = L_{ne}$ , we either obtain an elementary code word with the smallest weight $d_{\min }$ , or establish the lower bound of $L_{ne}$ on $d_{\min }$ . For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on $d_{\min }$ . We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $d_{\min }$ . Finding $d_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.