Good Code Sets from Complementary Pairs via Discrete Frequency Chips

It is shown that replacing the sinusoidal chip in Golay complementary code pairs by special classes of waveforms that satisfy two conditions, symmetry/anti-symmetry and quazi-orthogonality in the convolution sense, renders the complementary codes immune to frequency selective fading and also allows for concatenating them in time using one frequency band/channel. This results in a zero-sidelobe region around the mainlobe and an adjacent region of small cross-correlation sidelobes. The symmetry/anti-symmetry property results in the zero-sidelobe region on either side of the mainlobe, while quasi-orthogonality of the two chips keeps the adjacent region of cross-correlations small. Such codes are constructed using discrete frequency-coding waveforms (DFCW) based on linear frequency modulation (LFM) and piecewise LFM (PLFM) waveforms as chips for the complementary code pair, as they satisfy both the symmetry/anti-symmetry and quasi-orthogonality conditions. It is also shown that changing the slopes/chirp rates of the DFCW waveforms (based on LFM and PLFM waveforms) used as chips with the same complementary code pair results in good code sets with a zero-sidelobe region. It is also shown that a second good code set with a zero-sidelobe region could be constructed from the mates of the complementary code pair, while using the same DFCW waveforms as their chips. The cross-correlation between the two sets is shown to contain a zero-sidelobe region and an adjacent region of small cross-correlation sidelobes. Thus, the two sets are quasi-orthogonal and could be combined to form a good code set with twice the number of codes without affecting their cross-correlation properties. Or a better good code set with the same number codes could be constructed by choosing the best candidates form the two sets. Such code sets find utility in multiple input-multiple output (MIMO) radar applications

Good Random Matrices over Finite Fields

The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.Comment: 25 pages, publishe

Higher-order CIS codes

We introduce {\bf complementary information set codes} of higher-order. A binary linear code of length $tk$ and dimension $k$ is called a complementary information set code of order $t$ ($t$-CIS code for short) if it has $t$ pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a $t$-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order $3$ is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length $\le 12$ is given. Nonlinear codes better than linear codes are derived by taking binary images of $\Z_4$-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate $1/t$ it either provides $t$ disjoint information sets or proves that the code is not $t$-CIS. Using this algorithm, all optimal or best known $[tk, k]$ codes where $t=3, 4, \dots, 256$ and $1 \le k \le \lfloor 256/t \rfloor$ are shown to be $t$-CIS for all such $k$ and $t$, except for $t=3$ with $k=44$ and $t=4$ with $k=37$.Comment: 13 pages; 1 figur

On a Class of Optimal Nonbinary Linear Unequal-Error-Protection Codes for Two Sets of Messages

Several authors have addressed the problem of designing good linear unequal error protection (LUEP) codes. However, very little is known about good nonbinary LUEP codes. We present a class of optimal nonbinary LUEP codes for two different sets of messages. By combining t-error-correcting ReedSolomon (RS) codes and shortened nonbinary Hamming codes, we obtain nonbinary LUEP codes that protect one set of messages against any t or fewer symbol errors and the remaining set of messages against any single symbol error. For t ≥ 2, we show that these codes are optimal in the sense of achieving the Hamming lower bound on the number of redundant symbols of a nonbinary LUEP code with the same parameters

Peak Alignment of Gas Chromatography-Mass Spectrometry Data with Deep Learning

We present ChromAlignNet, a deep learning model for alignment of peaks in Gas Chromatography-Mass Spectrometry (GC-MS) data. In GC-MS data, a compound's retention time (RT) may not stay fixed across multiple chromatograms. To use GC-MS data for biomarker discovery requires alignment of identical analyte's RT from different samples. Current methods of alignment are all based on a set of formal, mathematical rules. We present a solution to GC-MS alignment using deep learning neural networks, which are more adept at complex, fuzzy data sets. We tested our model on several GC-MS data sets of various complexities and analysed the alignment results quantitatively. We show the model has very good performance (AUC $\sim 1$ for simple data sets and AUC $\sim 0.85$ for very complex data sets). Further, our model easily outperforms existing algorithms on complex data sets. Compared with existing methods, ChromAlignNet is very easy to use as it requires no user input of reference chromatograms and parameters. This method can easily be adapted to other similar data such as those from liquid chromatography. The source code is written in Python and available online