Complementary Codes Research Articles

SRIM and SCRIM factors of xn + 1 over finite fields and their applications

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SCRIM) factors of [Formula: see text] over finite fields have become of interest due to their rich algebraic structures and wide applications. In this paper, these notions are extended to factors of [Formula: see text] over finite fields. Characterization and enumeration of SRIM and SCRIM factors of [Formula: see text] over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the study of complementary negacyclic codes are discussed.

Classification of optimal quaternary Hermitian LCD codes of dimension $2$

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial.
 The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$. Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial.
 The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.

Construction of Z-Complementary Code Sets With Non-Power-of-Two Lengths Based on Generalized Boolean Functions

$Z$ -complementary code set (ZCCS) plays a major role in modern communication systems. It has applications in multi-carrier code-division multiple access (MC-CDMA) scheme, channel estimation in multiple-input and multiple-output (MIMO) systems, etc. In this letter, we propose a construction of $q$ -ary ZCCS of a length of the form $2^{m}+2$ using generalized Boolean functions (GBFs). The constructed ZCCS is optimal with respect to the theoretical upper bound. Based on the authors’ knowledge, all the existing constructions of ZCCSs based on GBFs are limited to the lengths of power-of-two only.

Fast Algebraic Immunity of Boolean Functions and LCD Codes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

CIS codes over ${\mathbb F}_4$

We study the complementary information set codes (for short, CIS codes) over ${\mathbb F}_4$. They are strongly connected to correlation-immune functions over ${\mathbb F}_4$. Also the class of CIS codes includes the self-dual codes. We find a construction method of CIS codes over ${\mathbb F}_4$ and a criterion for checking equivalence of CIS codes over ${\mathbb F}_4$. We complete the classification of all inequivalent CIS codes of length up to $8$ over ${\mathbb F}_4$.

A Near-Optimal and Low-Complex Joint Multiuser Detection for QCSS-MC-CDMA System

Joint maximum likelihood (ML) multiuser detection for quasi-synchronous multicarrier code division multiple access (MC-CDMA) system has an excessive computational complexity, and it increases exponentially as the number of users and bits in each of the modulation symbols increases. Recently, sphere decoding (SD) has been adopted for multiuser detection whose complexity is polynomial in nature with the number of users, and it does not rely on the number of bits in the modulation. In this article, first we provide a construction of large size quasi-complementary sequence sets (QCSSs), and then a low-complexity optimum SD algorithm is proposed to detect the multiuser communication in QCSS-based MC-CDMA (QCSS-MC-CDMA) system. The complexity analysis has shown that the proposed SD-based QCSS-MC-CDMA receiver for multiuser communication provides lower complexity than that of ML-based QCSS-MC-CDMA receiver. We also show that peak to average power ratio of the proposed system is lesser than that of complete complementary code (CCC) based MC-CDMA (CCC-MC-CDMA) system. The proposed system overcomes the fractional delay problem which is noticed for CCC-MC-CDMA receiver. The bit error rate performance of the proposed SD-based QCSS-MC-CDMA system is compared with maximum ratio combining based MC-CDMA, ML-based MC-CDMA, frequency domain equalization based MC-CDMA, and CCC-MC-CDMA receivers over Rayleigh frequency selective fading channels. The work is validated through Monte Carlo simulation results.

Complimentary code keying of spectral amplitude coding signals in optical buffering with increased capacity

Signal buffering services such as contention resolution and congestion avoidance are essential in optical packet switching networks. In this paper, an optical memory scheme based on spectral amplitude coding (SAC) and complementary code keying (CCK) is proposed to increase the buffer capacity. CCK is applied to packet buffering by selecting an available code set and encoding the payload bits with either an SAC signal or its complementarity. The capacity constraint is effectively released, as the usable codes for queuing packets are twice those for the conventional code-domain buffers. To minimize system costs by reducing the codec number, a shared structure based on an arrayed waveguide grating (AWG), which is capable of processing both the typical and complimentary coded signals simultaneously, is also investigated.

A Generalized Construction of Multiple Complete Complementary Codes and Asymptotically Optimal Aperiodic Quasi-Complementary Sequence Sets

In recent years, complete complementary codes (CCCs) and quasi-complementary sequence sets (QCSSs) have found many important applications in multi-carrier code-division multiple-access (MC-CDMA) systems for their good correlation properties. In this paper, we propose a generic construction of multiple sets of CCCs over $\mathbb {Z}_{N}$ , consisting of sequences of length $N$ , where $N\geq 3$ is an arbitrary odd integer. Interestingly, the maximum inter-set aperiodic cross-correlation magnitude of the proposed CCCs is upper bounded by $N$ . It turns out that the combination of the generated CCCs results in a new set of sequences to obtain asymptotically optimal and near-optimal aperiodic QCSSs. The proposed construction includes a recent optimal construction of QCSSs with prime length as a special case and leads to asymptotically optimal QCSSs with new flexible parameters.

Remark on subcodes of linear complementary dual codes

We show that any ternary Euclidean (resp. quaternary Hermitian) linear complementary dual [n,k] code contains a Euclidean (resp. Hermitian) linear complementary dual [n,k−1] subcode for 2≤k≤n. As a consequence, we derive a bound on the largest minimum weights among ternary Euclidean linear complementary dual codes and quaternary Hermitian linear complementary dual codes.

A Multisecret-Sharing Scheme Based on LCD Codes

Secret sharing is one of the most important cryptographic protocols. Secret sharing schemes (SSS) have been created to that end. This protocol requires a dealer and several participants. The dealer divides the secret into several pieces ( the shares), and one share is given to each participant. The secret can be recovered once a subset of the participants (a coalition) shares their information. In this paper, we present a new multisecret-sharing scheme inspired by Blakley’s method based on hyperplanes intersection but adapted to a coding theoretic situation. Unique recovery requires the use of linear complementary (LCD) codes, that is, codes in which intersection with their duals is trivial. For a given code length and dimension, our system allows dealing with larger secrets and more users than other code-based schemes.

The role of adaptation in generating monotonic rate codes in auditory cortex.

In primary auditory cortex, slowly repeated acoustic events are represented temporally by the stimulus-locked activity of single neurons. Single-unit studies in awake marmosets (Callithrix jacchus) have shown that a sub-population of these neurons also monotonically increase or decrease their average discharge rate during stimulus presentation for higher repetition rates. Building on a computational single-neuron model that generates stimulus-locked responses with stimulus evoked excitation followed by strong inhibition, we find that stimulus-evoked short-term depression is sufficient to produce synchronized monotonic positive and negative responses to slowly repeated stimuli. By exploring model robustness and comparing it to other models for adaptation to such stimuli, we conclude that short-term depression best explains our observations in single-unit recordings in awake marmosets. Together, our results show how a simple biophysical mechanism in single neurons can generate complementary neural codes for acoustic stimuli.

Multi-frame motion deblurring using coded exposure imaging with complementary fluttering sequences

Motion blurring is one of the most common but unpleasant phenomenons along with the camera imaging process, resulting in image resolution degradation. Coded exposure technique is an effective computational imaging approach to handle with this problem by introducing extra shutter control within the exposure time and help the deblurring problem become well-posed. In this paper, a multi-frame motion deblurring framework is presented using coded exposure imaging with complementary fluttering sequences. A set of complementary sequences are well designed to compensate for the spatial losses due to motion blurring in each frame based on the complementary theory of Golay matrix pair. Hadamard matrix is served as the Golay sequence pair to generate complementary fluttering sequences to capture multi-frame images for the following deblurring processing. The joint modulation transfer function (MTF) of the complementary code sequences has a more flat spectrum curve with smaller variance value and higher MTF value. For multi-frame deblurring implementation, a unified multi-image deblurring framework is performed by introducing Total Variation (TV) regularization prior with each selected code sequence in the complementary set integrated in the energy function. A real imaging system is built and various groups of synthetic and real-world imaging experiments are carried out to verify the effectiveness of the method. Experimental results demonstrate that the deblurrring images obtained by the proposed approach are of high quality with improved objective image evaluation values and better negative artifacts suppression.

Light-Responsive Colloidal Crystals Engineered with DNA.

A novel method for synthesizing and photopatterning colloidal crystals via light-responsive DNA is developed. These crystals are composed of 10-30 nm gold nanoparticles interconnected with azobenzene-modified DNA strands. The photoisomerization of the azobenzene molecules leads to reversible assembly and disassembly of the base-centered cubic (bcc) and face-centered cubic (fcc) crystalline nanoparticle lattices. In addition, UV light is used as a trigger to selectively remove nanoparticles on centimeter-scale thin films of colloidal crystals, allowing them to be photopatterned into preconceived shapes. The design of the azobenzene-modified linking DNA is critical and involves complementary strands, with azobenzene moieties deliberately staggered between the bases that define the complementary code. This results in a tunable wavelength-dependent melting temperature (Tm ) window (4.5-15 °C) and one suitable for affecting the desired transformations. In addition to the isomeric state of the azobenzene groups, the size of the particles can be used to modulate the Tm window over which these structures are light-responsive.

Multiple Complete Complementary Codes With Inter-Set Zero Cross-Correlation Zone

Complete complementary codes (CCCs) have found a number of practical applications in wireless communications owing to their perfect aperiodic auto-correlation and cross-correlation properties. In particular, an important application of CCCs is in interference-free multi-carrier code-division multiple-access (MC-CDMA) systems. The available number of complementary codes determines the system capacity. To enlarge the system capacity, this paper focuses on constructing multiple CCCs with inter-set zero cross-correlation zone (ZCCZ) property based on generalized Boolean functions (GBFs). To the best of our knowledge, such multiple CCCs have not been reported in the literature. The obtained multiple CCCs can provide advantages in applications to MC-CDMA systems with multiple cells. As a byproduct, optimal aperiodic Z-complementary code sets (ZCCSs) with low peak-to-mean envelope power ratio (PMEPR) can be derived by combining these CCCs.

Two-Dimensional $Z$-Complementary Array Code Sets Based on Matrices of Generating Polynomials

The unprecedented elevation of wireless technologies produces the ever-increasing requirement for higher-dimensional array sets with preferable correlation properties and more flexible parameters. In this paper, we investigate new two-dimensional $Z$ -complementary array code sets (2D-ZCACSs) with unimodular arrays, which are more preferable in ultra wideband communication system to improve its performance than 2D mutually orthogonal complementary array set. We first introduce a new idea of 2D $Z$ -paraunitary (2D-ZPU) matrices by extending our previous concept of 1D-ZPU matrices. Then, we show that there exists an equivalence between a 2D-ZCACS and 2D-ZPU matrix when the former is expressed as a matrix with 2D generating polynomial entries. In the 2D generating matrix, each column corresponds to a 2D $Z$ -complementary array code and two distinct columns correspond to 2D $Z$ -complementary array mates. Finally, we propose a new construction method of unimodular 2D-ZCACSs with the aid of higher-dimensional $z$ -transforms and the theory of ZPU matrices.

Range-Doppler Sidelobe Suppression for Pulsed Radar Based on Golay Complementary Codes

To relieve the interference caused by range-Doppler sidelobes in pulsed radars, we propose a new method to construct Doppler resilient complementary waveforms based on Golay codes. We design both the transmit pulse train and the receive pulse weights, so that the similarity between the pulse weights and a given window function is maximized and the constraints on Doppler null points and energy are met. That is summarized as a two-way partitioning problem, and then solved by semidefinite programming and randomization techniques. The novel waveform thus obtained has its range sidelobe outright suppressed in multiple and flexibly-adjustable Doppler zones, and performs well in Doppler sidelobe suppression, Doppler resolution and SNR. It shows great promise in detecting slightly-moving weak targets with the existence of dense interference.

Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems

In the paper, a deoxyribonucleic acid (DNA) extension code with 3-bit binary streams is proposed to encrypt the downlink data for orthogonal frequency division multiplexing passive optical network (OFDM-PON). It has 8 bases to make up 4 pairs of complementary codes. And it can obtain 384 matching rules, which greatly improves the randomness of matching. Here, two DNA addition operation rules are also proposed to encrypt the data. DNA extension rules can reduce half coding operations. Three 1-dimensional (1-D) chaotic systems are used to encrypt the code and control the rules. The encryption method based on the uplink streams from optical network units (ONUs) makes the security of downlink signals not just depending on the security of chaotic systems. Finally, a 22.06 Gb/s DNA extension code encryption signal is transmitted through a back-to-back (BTB) system and a 25-km standard single-mode fiber (SSMF). The proposed method not only improves the security but also reduces the computational complexity. The experimental results show that the proposed method has the ability to resist optical channel response and fiber nonlinearity, which is a promising candidate for solving the security enhancement in access networks.

New Optimal $Z$-Complementary Code Sets Based on Generalized Paraunitary Matrices

The concept of paraunitary (PU) matrices arose in the early 1990 s in the study of multi-rate filter banks. These matrices have found wide applications in cryptography, digital signal processing, and wireless communications. Existing PU matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice. Motivated by this, for the first time, we introduce a novel concept, called $Z$ -paraunitary (ZPU) matrix, whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value. We show that there exists an equivalence between a ZPU matrix and a $Z$ -complementary code set when the latter is expressed as a matrix with polynomial entries. Furthermore, we investigate some important properties of ZPU matrices, which are useful for the extension of matrix sizes and sequence lengths. We propose a new construction framework for optimal ZPU matrices which includes existing PU matrices as special cases. Finally, we give the enumeration formula for the constructed $Z$ -complementary sequences from the optimal ZPU matrices.

On [formula omitted]-additive complementary dual codes and related LCD codes

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive subgroups, rather than linear as they are not vector spaces over some finite field. We study the structure of these codes and we use the canonical Gray map from this space to the Hamming space to construct binary LCD codes in certain cases. We give examples of such binary LCD codes which are distance-optimal, i.e., they have the largest minimum distance among all binary LCD codes with the same length and dimension.

Hadamard-Type Matrices on Finite Fields and Complete Complementary Codes

Hadamard matrix is defined as a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. In this work, we consider the similar matrix on finite field GF(p) where p is an odd prime. In such a matrix, every component is one of the integers on GF(p)\{0}, that is, {1,2,...,p-1}. Any additions and multiplications should be executed under modulo p. In this paper, a method to generate such matrices is proposed. In addition, the paper includes the applications to generate n-shift orthogonal sequences and complete complementary codes. The generated complete complementary code is a family of multi-valued sequences on GF(p)\{0}, where the number of sequence sets, the number of sequences in each sequence set and the sequence length depend on the various divisors of p-1. Such complete complementary codes with various parameters have not been proposed in previous studies.