arXiv subjects
Publications and source records attributed to Martianus Frederic Ezerman.
We construct qubit stabilizer codes with parameters $[[81, 0, 20]]$ and $[[94, 0, 22]]$ for the first time. We use symplectic self-dual additive codes over $\mathbb{F}_4$ built by modifying the adjacency matrices of suitable metacirculant graphs found by a randomized search procedure.
We use symplectic self-dual additive codes over $\mathbb{F}_4$ obtained from metacirculant graphs to construct, for the first time, $[[\ell, 0, d ]]$ qubit codes with parameters $(\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,21)\}$. Secondary constructions applied to the qubit codes result in many qubit codes that perform better than the previous best-known.
Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other.
We study how to generate binary de Bruijn sequences efficiently from the class of simple linear feedback shift registers with feedback function $f(x_0, x_1, \ldots, x_{n-1}) = x_0 + x_1 + x_{n-1}$ for $n \geq 3$, using the cycle joining method. Based on the properties of this class of LFSRs, we propose two new generic successor rules, each of which produces at least $2^{n-3}$ de Bruijn sequences. These two classes build upon a framework proposed by Gabric, Sawada, Williams and Wong in Discrete Mathematics vol. 341, no. 11, pp. 2977--2987, November 2018. Here we introduce new useful choices for the uniquely determined state in each cycle to devise valid successor rules. These choices significantly increase the number of de Bruijn sequences that can be generated. In each class, the next bit costs $O(n)$ time and $O(n)$ space for a fixed $n$.
Holographic representations of data enable distributed storage with progressive refinement when the stored packets of data are made available in any arbitrary order. In this paper, we propose and test patch-based transform coding holographic sensing of image data. Our proposal is optimized for progressive recovery under random order of retrieval of the stored data. The coding of the image patches relies on the design of distributed projections ensuring best image recovery, in terms of the $\ell_2$ norm, at each retrieval stage. The performance depends only on the number of data packets that has been retrieved thus far. Several possible options to enhance the quality of the recovery while changing the size and number of data packets are discussed and tested. This leads us to examine several interesting bit-allocation and rate-distortion trade offs, highlighted for a set of natural images with ensemble estimated statistical properties.
The article surveys quantum error control, focusing on quantum stabilizer codes, stressing on the how to use classical codes to design good quantum codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC Press.
We solve an open question in code-based cryptography by introducing two provably secure group signature schemes from code-based assumptions. Our basic scheme satisfies the CPA-anonymity and traceability requirements in the random oracle model, assuming the hardness of the McEliece problem, the Learning Parity with Noise problem, and a variant of the Syndrome Decoding problem. The construction produces smaller key and signature sizes than the previous group signature schemes from lattices, as long as the cardinality of the underlying group does not exceed $2^{24}$, which is roughly comparable to the current population of the Netherlands. We develop the basic scheme further to achieve the strongest anonymity notion, i.e., CCA-anonymity, with a small overhead in terms of efficiency. The feasibility of two proposed schemes is supported by implementation results. Our two schemes are the first in their respective classes of provably secure groups signature schemes. Additionally, the techniques introduced in this work might be of independent interest. These are a new verifiable encryption protocol for the randomized McEliece encryption and a novel approach to design formal security reductions from the Syndrome Decoding problem.
We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite (GPO) Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most previously-known greedy algorithms that generate binary de Bruijn sequences are particular cases of our new algorithm.
We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over $\mathbb{F}_4$ and $\mathbb{F}_9$ to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.
New lower bounds on the minimum distance of quasi-twisted codes over finite fields are proposed. They are based on spectral analysis and eigenvalues of polynomial matrices. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a manner similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes.
The focus of this work is to show how to combine Zech's logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences of any order. A basic implementation is supplied as a proof-of-concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. Certificates of star or almost-star spanning trees conveniently handle large order cases. The characterization of conjugate pairs via Zech's logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified $m$-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. We show how to integrate an analytic tool, attributed to Fryers, in the process. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.
This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize $\mathbb{F}_4R$-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over $\mathbb{F}_4$ are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of $R$-skew cyclic codes which are reversible complement.
Holographic representations of data encode information in packets of equal importance that enable progressive recovery. The quality of recovered data improves as more and more packets become available. This progressive recovery of the information is independent of the order in which packets become available. Such representations are ideally suited for distributed storage and for the transmission of data packets over networks with unpredictable delays and or erasures. Several methods for holographic representations of signals and images have been proposed over the years and multiple description information theory also deals with such representations. Surprisingly, however, these methods had not been considered in the classical framework of optimal least-squares estimation theory, until very recently. We develop a least-squares approach to the design of holographic representation for stochastic data vectors, relying on the framework widely used in modeling signals and images.
We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial $f(x)$. We study in detail the cycle structure of the set $Ω(f(x))$ that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.
We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.
We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial $f(x)=p(x)q(x)$ where $p(x)$ and $q(x)$ are distinct irreducible polynomials in $\F_2[x]$. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to find all conjugate pairs shared by any pair of cycles are given. The process explicitly determines the edges and their labels in the adjacency graph. The results are then combined with the cycle joining method to efficiently construct a new class of de Bruijn sequences. An estimate of the number of resulting sequences is given. In some cases, using cyclotomic numbers, we can determine the number exactly.
A recent study by one of the authors has demonstrated the importance of profile vectors in DNA-based data storage. We provide exact values and lower bounds on the number of profile vectors for finite values of alphabet size $q$, read length $\ell$, and word length $n$.Consequently, we demonstrate that for $q\ge 2$ and $n\le q^{\ell/2-1}$, the number of profile vectors is at least $q^{κn}$ with $κ$ very close to one.In addition to enumeration results, we provide a set of efficient encoding and decoding algorithms for each of two particular families of profile vectors.
The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the product of the reversals of $A$ and of $B$. A pair $(A,B)$ of positive integers is said to be polynomial if the product $A \times B$ can be performed without carry. In this paper, we use polynomial pairs in constructing and in studying the properties of palindromic pairs. It is shown that polynomial pairs are always palindromic. It is further conjectured that, provided that neither $A$ nor $B$ is itself a palindrome, all palindromic pairs are polynomial. A connection is made with classical topics in recreational mathematics such as reversal multiplication, palindromic squares, and repunits.