Prime Length Research Articles

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Statistical Validation of a Physical Prime Random Number Generator Based on Quantum Noise

Random prime numbers are an essential resource for many asymmetric cryptographic protocols. However, despite the emerging popularity of quantum random number generators (QRNGs) as sources of secure randomness, physical prime number generators have not yet been explored. In this work, we experimentally implement and characterize a vacuum-based probabilistic prime number generation scheme with an error probability of 3.5×10−15. By removing the quantum source (QS), an additional scheme based on electronic noise is derived, and a comparative analysis for increasing prime lengths is made. We observed that the QS significantly outperforms the classical scheme for small prime generation, where increases up to 585.0% in the diversity of unique primes obtained are seen. Moreover, we propose a length-agnostic statistical test for prime number sequences and apply it to the output of the uniformized randomness source, which was successful in revealing underlying biases in the output prime distributions. The resultant sequences were subsequently submitted to the NIST statistical test suite, where the quantum and classical sources passed, respectively, 86.96% and 45.34% of the total test set applied.

Weight enumerators of all cubic-primitive irreducible cyclic codes of odd prime power length

Let p and q be odd primes and q be a cubic primitive modulo pu for some positive integer u. In this paper, we prove that the solutions of some Diophantine equations provide the weight enumerators of some cubic primitive irreducible cyclic codes of prime length. Bounds on the minimum distances of these codes are also given.

Certain diagonal equations and conflict-avoiding codes of prime lengths

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g2Xℓ+gYℓ+1=0 over the finite field Fp for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field Fq of Fp. We show that for q greater than a lower bound of the order of magnitude O(ℓ2) there exists a generator g of Fq× such that the equation in question is solvable over Fq. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

Threats and the conservative shift in public opinion: comparative causal mediation analyses from Turkey

ABSTRACT How do mortality reminders and aroused mortality fear affect people’s conservatism? Threats-politics studies assume that life-threatening dangers, fear of death, and conservatism go together, overlooking the distinction between threat exposure and mortality fear arousal. Further, these studies commonly fixate on the conventional threat of terrorism and predominantly employ samples from Western countries. Complementing past research, I use three survey experiments in Turkey to investigate the relationship between a broad array of threats (terrorism, health risks, moral hazards, and daily thoughts of death’s inevitability), mortality fear induced by the threats, and self-rated conservative orientation in a causal mediation analysis framework, using the terror management theory and non-representative Facebook samples (2018, 2020). The results reveal that the respondents receiving threatening stimuli (particularly terrorism mortality reminder) become less conservative as the mortality fear elevates. The length of mortality priming and the broader study context influence this indirect effect’s significance and replicability.

Abstract 11527: Regional Variation in Cardiovascular Genes Enables a Tractable Genome Editing Strategy

Recent rise in the number and therapeutic potential of genome engineering technologies has generated excitement in cardiovascular genetics. One significant barrier to their implementation is costly and time consuming reagent development for novel unique variants. Prior data have illuminated the potential for regional clustering of disease-causing genetic variants in known and potential novel functional protein domains. We hypothesized that most cardiovascular disease-relevant genes in ClinVar would display regional variant clustering, and that multiple variants within a regional hotspot could be targeted with limited prime editing reagents. We collated 2,471 high confidence pathogenic or likely pathogenic (P/LP) missense and truncating variants from 111 cardiovascular disease genes. We then defined a regional clustering index (RCI), the percent of P/LP variants in a given gene located within a pre-specified window around each variant. At a window size commensurate with maximally reported prime editing extension length, 78bp, we found that missense variants displayed a higher mean RCI than truncating variants. We next identified genes particularly attractive for pathogenic hotspot-targeted prime editing with at least 20 P/LP variants and observed that the mean RCI in missense variants remained higher than for truncating variants and for rare variants observed in the same genes in gnomAD (Mean±SD RCI 78bp : P/LP ClinVar Missense=5.2±5.9%; P/LP ClinVar Truncating=2.1±3.2%, gnomAD Missense=2±3.2%, gnomAD Truncating=1.1±2.4%. p<2.2e-16; ANOVA with post hoc Bonferroni correction). Further, we tested the feasibility of prime editing for multiple variants in a single hotspot in KCNH2, a gene with a high mean missense variant RCI. Sixty variants were induced in this hotspot in HEK293 cells with CRISPR-X. Correction of these variants was attempted with prime editing using two overlapping prime editing guide RNA extensions. The mean prime editing efficiency across CRISPR-X-enriched variants within this hotspot was 57±27%, including 3 P/LP variants. These data underscore the utility of pathogenic variant hotspots in the diagnosis and treatment of inherited cardiovascular disease.

Analysis of Generating Functions for Special Words and Numbers and Algorithms for Computation

Our aim is to construct and compute efficient generating functions enumerating the k-ary Lyndon words having prime number length which arise in many branches of mathematics and computer science. We prove that these generating functions coincide with the Apostol–Bernoulli numbers and their interpolation functions and obtain other forms of these generating functions including not only the Frobenius–Euler numbers, but also the Fubini type numbers. Moreover, we derive some identities, relations and combinatorial sums including the numbers of the k-ary Lyndon words, the Bernoulli numbers and polynomials, the Stirling numbers and falling factorials. Using these generating functions and recurrence relation for the Apostol–Bernoulli numbers, we give two algorithms to compute these generating functions. Using these algorithms, we compute some infinite series formulas including the number of the k-ary Lyndon words on some special classes of primes with the purpose of providing some numerical evaluations about these generating functions. In addition, we approximate these generating functions by the rational functions of the Apostol–Bernoulli numbers to show that the complexity of the aforementioned algorithms may be decreased by means of approximation method which are illustrated by some numerical evaluations with their plots for varying prime numbers. Finally, using Bell polynomials (i.e., exponential functions) approach to the numbers of Lyndon words, we construct the exponential generating functions for the numbers of Lyndon words. Finally, we define a new family of special numbers related to these special words and investigate some of their fundamental properties.

Dimension Theory in Iterated Local Skew Power Series Rings

Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull and global dimensions, obtaining lower bounds to complement the upper bounds obtained by Wang. In fact, we show that many common such rings obey a stronger property, which we call triangularity, and which allows us also to calculate their classical Krull dimension (prime length). Finally, we correct an error in the literature regarding the associated graded rings of general iterated skew power series rings, but show that triangularity is enough to recover this result.

Oriented expressions of graph properties

Several graph properties are characterized as a class of graphs that admit an orientation avoiding a finite set of oriented structures. For instance, if Fk is the set of homomorphic images of the directed path on k+1 vertices, then a graph is k-colourable if and only if it admits an orientation with no induced oriented subgraph in Fk. There is a fundamental question underlying this kind of characterizations: Given a graph property P, is there a finite set of oriented graphs F such that a graph belongs to P if and only if it admits an orientation with no induced oriented subgraph in F? We address this question by exhibiting some necessary conditions upon certain graph classes for them to admit such a characterization. Consequently, we exhibit an uncountable family of hereditary classes for which no such finite set exists. In particular, the class of graphs with no holes of prime length belongs to this family.

Duality of constacyclic codes of prime power length over the finite non-commutative chain ring [formula omitted

Let R=Fpm[u,θ]〈u2〉 where θ is an automorphism of Fpm but it is not the identity. The list of (left and right) α-constacyclic codes of length ps over R is provided where θ(α)=α. The self-dual α-constacyclic codes are given. In addition, we derive the condition of LCD codes for those codes. For the remaining result, the condition of self-orthogonal left α-constacyclic codes is also obtained.

Periodic On–Off Keying Codes for Laser Radar Incoherent Pulse Compression

Pulsed laser systems for long range target detection process multiple pulses to increase signal-to-noise ratio SNR from that of a single pulse. Unipolar <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> – <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> keying (OOK) periodic pulse modulation codes are appropriate for an incoherent system. Mismatched filters for cross-correlation with codes are solutions to exactly determined linear equations, so all zero sidelobes are realized for any nonsingular set of equations and codes that are not all zeros. Therefore, maximum SNR is proposed as the criterion for code-filter pair adoption at a given code length. The fast solution of the mismatched filtering circulant equations allows all codes for a set of balancing values [differences of numbers of <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> (ones) and <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> (zeros)] to be tested for odd code lengths up through 33. M-sequences and Legendre codes of prime lengths one less than a multiple of four and an excess of one more <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> than <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> lead to optimal SNR code-filter pairs. These Legendre codes are suggested for longer codes since they are easily constructed. Code-filter pairs that optimize SNR for a given code length and balancing values of ±1, ±3, ±5, and ±7 are reported. For a balancing value of –1, a new set of codes with corresponding two-value filters is found.

Accelerating Stochastic Computing Using Deterministic Halton Sequences

Deterministic approaches have recently been developed for accurate computation in stochastic computing (SC). They, however, suffer a long operation time. Fortunately, for applications that do not require completely accurate processing results, such as image processing, the time can significantly be reduced due to the better progressive precision in the bit-streams generated by these approaches. That means a computation can be terminated in time when its output accuracy is acceptable. Due to the fast convergence property of low-discrepancy sequences, we propose three deterministic Halton sequence (DHS)-based stochastic number generators (SNGs) for the first time by using, respectively, prime length, rotation, and clock division for accelerating computation in SC. Experimental results show that the proposed designs are more efficient than their counterparts. For multiplication, the proposed DHS-based designs perform up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$32\times $ </tex-math></inline-formula> faster than prior designs for a mean error of 0.1%. The speedup reaches <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$128\times $ </tex-math></inline-formula> for an edge detection algorithm. Three stochastic circuits are then designed by using the proposed DHS-based SNGs for the Bernsen binarization algorithm, which lead to more accurate results than existing designs at the same bit-stream length. Finally, the proposed designs show an excellent fault-tolerance against bit flipping errors.

How to improve seed germination with green nanopriming

The ability to improve the germination performance of aged or dormant seeds is significant in agriculture. Seed nanopriming is one potentially effective strategy. The method of synthesising nanoparticles (NPs) is generally chemical reduction. However, most of the reducing agents are highly toxic chemicals. Plant substitutes can replace such toxic reductants because of their environmental friendliness and hence green synthesis of NPs has arisen as an emerging technology. The technology of seed priming with green synthetic NPs is called green nanopriming. This review introduces the concept of nanopriming, which has attracted research interest in recent years due to advances in nanotechnology. The basis of green nanopriming, the method for green synthesis of NPs, conditions of green nanopriming (temperature; concentration of NPs; length of priming; nature of NPs) are elaborated. Further, the effects on seeds and the growth of seedlings are also summarised. This technology can promote germination of seeds effectively without poisoning seedlings. Therefore, green nanopriming has huge potential for enhancing crop productivity.

Most permutations power to a cycle of small prime length

AbstractWe prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

Performance and meat quality of broilers fed different levels of alpha-tocopherol

This study aimed to evaluate the performance and meat quality of broilers fed different levels of alpha-tocopherol. A total of 240 one-day-old Cobb 500® male broiler chicks were reared until day 7 of life, as recommended by the commercial line, and were fed at only 50% of the alpha-tocopherol requirement. At day 8, the chicks were weighed, homogenized, and allotted in a completely randomized design with four treatments (50, 100, 150, and 200% alpha-tocopherol requirement) and six replicates of ten chickens per experimental unit. Several variables were evaluated: weight gain, feed intake, feed conversion, carcass yield and prime cuts (drumstick, thigh, and breast), edible viscera (heart, liver, gizzard), immune organs (cloacal bursa and spleen), intestine weight and length, abdominal fat, breast meat color (lightness, redness, and yellowness), pH, cooking loss, and shear force of the breast meat. Dietary alpha-tocopherol levels did not influence broiler performance, carcass yield, and prime cuts at 42 days of age. The edible viscera, immune organs, the length of the small intestine, and the abdominal fat, as well as the meat color (lightness, redness, and yellowness), pH, cooking loss, and shear force of the breast meat, were not influenced. However, the small intestine yield increased linearly with increasing levels of alpha-tocopherol. The dietary levels of alphatocopherol did not influence the performance and meat quality of broilers from 8 to 42 days of age.

Qadratic residue codes over 𝔽p + u𝔽p + v𝔽p∗

In this paper, we study quadratic residue (QR) codes of prime length [Formula: see text] over the ring [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length [Formula: see text] over [Formula: see text], we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good [Formula: see text]-ary codes as Gray images of QR codes over [Formula: see text] by considering the case when [Formula: see text] is an odd prime.

Roulette games and depths of words over finite commutative rings

In this paper, we propose three new turn-based two player roulette games and provide positional winning strategies for these games in terms of depths of words over finite commutative rings with unity. We further discuss the feasibility of these winning strategies by studying depths of codewords of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over a finite commutative chain ring $${\mathcal {R}},$$ where $$\alpha $$ is a non-zero element of the Teichmuller set of $${\mathcal {R}},$$ $$\gamma $$ is a generator of the maximal ideal of $${\mathcal {R}}$$ and $$\beta $$ is a unit in $${\mathcal {R}}.$$ As a consequence, we explicitly determine depth distributions of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over $${\mathcal {R}}$$ .

Shared-ACS Binary Code Pair Search Exploiting Coupled Symparity

Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist code pairs for which the mechanism behind shared autocorrelation is not so simple; these are called shared-autocorrelation (or shared-ACS) binary code pairs. They are useful for, among other things, constructing binary and quad-phase Golay pairs of longer lengths. We define a symparity vector for any given binary code. For every shared-ACS binary code pairs up to length 39, the two code have the same symparity, or coupled symparity. Conditions are identified for which shared-ACS binary code pairs have coupled symparity. The coupled-symparity condition is used to develop an algorithm to find shared-ACS binary code pairs, with reduced complexity relative to more brute-force exhaustive searches. Although the algorithm assumes extra structure in the shared-ACS code pairs, it rediscovered all the sets of binary shared-ACS code pairs for lengths up to 20. It is then run for lengths 40 to 44, returning all the shared-ACS binary code pairs for each length. In particular, the algorithm results establish that there exist shared-ACS code pairs for the prime lengths 41 and 43.

Counting the decimation classes of binary vectors with relatively prime length and density

We present a method for determining the number of decimation classes of density $$\delta $$ binary vectors indexed by a finite abelian group G of size $$\ell $$ and exponent $${\ell ^*}$$ such that $$\delta $$ is relatively prime to $${\ell ^*}$$ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of $${{\mathbb {Z}}_{{\ell ^*}}^{\times }}$$ . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by $${\mathbb {Z}}_{\ell }$$ of odd length $$\ell $$ and density $$(\ell +1)/2$$ greatly increased the number of $$\ell $$ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.

A New Formulation of a Criterion for the Minimal Logarithmic Growth Rate

We obtain a new formulation of a criterion for the minimal logarithmic growth rate for an arbitrary finite set with a given set of operations. It turns out that a finite set with operations has the minimal logarithmic growth rate if and only if the set of operations is not entirely contained in any of the precomplete (maximal) classes other than the classes preserving subsets and the classes of functions that preserve predicates given by permutations decomposable into cycles of the same prime length.