Number Of Codewords Research Articles

On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.

Entropy-Optimized Deep Weighted Product Quantization for Image Retrieval.

Hashing and quantization have greatly succeeded by benefiting from deep learning for large-scale image retrieval. Recently, deep product quantization methods have attracted wide attention. However, representation capability of codewords needs to be further improved. Moreover, since the number of codewords in the codebook depends on experience, representation capability of codewords is usually imbalanced, which leads to redundancy or insufficiency of codewords and reduces retrieval performance. Therefore, in this paper, we propose a novel deep product quantization method, named Entropy Optimized deep Weighted Product Quantization (EOWPQ), which not only encodes samples into the weighted codewords in a new flexible manner but also balances the codeword assignment, improving while balancing representation capability of codewords. Specifically, we encode samples using the linear weighted sum of codewords instead of a single codeword as traditionally. Meanwhile, we establish the linear relationship between the weighted codewords and semantic labels, which effectively maintains semantic information of codewords. Moreover, in order to balance the codeword assignment, that is, avoiding some codewords representing most samples or some codewords representing very few samples, we maximize the entropy of the coding probability distribution and obtain the optimal coding probability distribution of samples by utilizing optimal transport theory, which achieves the optimal assignment of codewords and balances representation capability of codewords. The experimental results on three benchmark datasets show that EOWPQ can achieve better retrieval performance and also show the improvement of representation capability of codewords and the balance of codeword assignment.

Information Bottleneck-Inspired Type Based Multiple Access for Remote Estimation in IoT Systems

Type-based multiple access (TBMA) is a semantics-aware multiple access protocol for remote inference. In TBMA, codewords are reused across transmitting sensors, with each codeword being assigned to a different observation value. Existing TBMA protocols are based on fixed shared codebooks and on conventional maximum-likelihood or Bayesian decoders, which require knowledge of the distributions of observations and channels. In this letter, we propose a novel design principle for TBMA based on the information bottleneck (IB). In the proposed IB-TBMA protocol, the shared codebook is jointly optimized with a decoder based on artificial neural networks (ANNs), so as to adapt to source, observations, and channel statistics based on data only. We also introduce the Compressed IB-TBMA (CIB-TBMA) protocol, which improves IB-TBMA by enabling a reduction in the number of codewords via an IB-inspired clustering phase. Numerical results demonstrate the importance of a joint design of codebook and neural decoder, and validate the benefits of codebook compression.

Half-Period Gray-Level Coding Strategy for Absolute Phase Retrieval

N-ary gray-level (nGL) coding strategy is an effective method for absolute phase retrieval in the fringe projection technique. However, the conventional nGL method contains many unwrapping errors at the boundaries of codewords. In addition, the number of codewords is limited in only one pattern. Consequently, this paper proposes a new gray-level coding method based on half-period coding, which can improve both these two deficiencies. Specifically, we embed every period with a 2-bit codeword, instead of a 1-bit codeword. Then, special correction and decoding methods are proposed to correct the codewords and calculate the fringe orders, respectively. The proposed method can generate n2 codewords with n gray levels in one pattern. Moreover, this method is insensitive to moderate image blurring. Various experiments demonstrate the robustness and effectiveness of the proposed strategy.

Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Improved dual-frequency phase-coding fringe projection method for 3D shape measurement

The phase-coding method is useful in 3D shape measurement due to its low sensitivity to surface contrast and camera noise. However, as the number of fringe periods increases, the wrapped phase recovered from the sinusoidal phase-shifting patterns is always misaligned with the fringe order calculated by the stair-coding patterns. To solve this, an improved dual-frequency phase-coding fringe projection method is designed to realize absolute phase retrieval. Specifically, the auxiliary fringe order determined by phase-shifting is used to get the absolute phase, which can eliminate fringe-order errors. Moreover, to increase speed and the number of codewords, this paper presents a scheme that only requires seven patterns, including a uniform gray image, two high-frequency sinusoidal fringe patterns, two low-frequency sinusoidal fringe patterns and two phase-coding fringe patterns. Therefore, this method requires fewer patterns to realize high-resolution measurement than existing methods. The success of the proposed method can be proved by experiments.

Period-Wise Phase Unwrapping Method With Two Gray Level Coding Patterns

Fringe projection profilometry is a widely used three-dimensional (3D) optical measurement technique. However, it is quite challenging to quickly obtain the 3D information of objects. Minimizing the number of patterns for phase unwrapping is an efficient way to address the above problem, and many methods have been proposed. This paper reports a period-wise phase unwrapping method with only two gray level coding patterns. Due to a hybrid coding strategy implemented, the number of patterns used in the proposed method reduces by one in comparison with conventional gray level method when generating the same number of codewords. A robust decoding scheme is also developed to determine the fringe order, and its accuracy is guaranteed by using the self-correcting method to remove the phase errors. Some experiments have been conducted, and their results confirm that the success of the proposed method.

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

Analysis of error correction performance for error correcting codes is very important when using such codes in digital communication systems. At medium-to-high signal-to-noise ratios, the distance spectrum of the error correcting code represents a good indicator for the error correction performance of the code. It is desired that the minimum distance of the code is as large as possible and that the corresponding multiplicity (i.e. the number of codewords having the weight equal to the minimum distance) is as small as possible. If we know an upper bound of the minimum distance of the code, then we have a good indication about the capabilities and the limitations of the code. One of the classes of the error correcting codes with the best performance is that of turbo codes. For such codes, establishing upper bounds on the minimum distance is challenging because it depends on the interleaver component of the turbo code. In this paper we consider turbo codes with component convolutional codes as in the Long Term Evolution standard. The interleaver lengths are of the form $$16 \varPsi $$ or $$48 \varPsi $$ , with $$ \varPsi $$ a product of different prime numbers greater than three. The first achievement in the paper is that for these interleaver lengths, we show that cubic permutation polynomials (CPP), with some constraints on the coefficients, when $$3 \not \mid (p_i-1)$$ for a prime $$p_i > 3$$ , always have a true inverse CPP. The most accurate upper bounds on the minimum distance for turbo codes are achieved by identifying bit information sequences leading to a certain weight of the corresponding turbo-codeword. In this paper we have indentified such bit information sequences by means of the full range dual impulse method to estimate the weight of the turbo-codewords. For the previously mentioned turbo codes and CPP interleavers, we show that the minimum distance is upper bounded by the values of 38, 36, and 28, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for quadratic PP (QPP) interleavers, the upper bound of the minimum distance is equal to 38. Several examples show that $$d_{min}$$ -optimal CPP interleavers are better than $$d_{min}$$ -optimal QPP interleavers because the multiplicities corresponding to the minimum distances for CPPs are about a half of those for QPPs. A theoretical explanation in terms of nonlinearity degrees for this result is given for all considered interleaver lengths and for the class of CPPs for which the upper bound is equal to 38.

3D measurement method based on S-shaped segmental phase encoding

Phase-based encoding has good immunity and robustness due to its insensitivity to ambient light, camera noise, and surface reflectance of measured objects. However, the number of code words is limited using traditional phase encoding method. In order to solve this problem, this paper proposes a new approach named S-shaped segmental phase encoding for 3D measurement. Specifically, by using a segmentation mechanism, the proposed method can, firstly, expand the range of code words whose number is limited using traditional phase encoding method. Secondly, the measurement accuracy can be improved. In this paper, Gray code is used to encode the phase encoding segment number, such that the fringe order judgment accuracy can be guaranteed. The experimental results demonstrated more accurate and efficient 3D shape measurement than the traditional methods, especially regarding the measurement of complex and isolated objects.

A Security Enhanced Encryption Scheme and Evaluation of Its Cryptographic Security.

An approach for security enhancement of a class of encryption schemes is pointed out and its security is analyzed. The approach is based on certain results of coding and information theory regarding communication channels with erasures and deletion errors. In the security enhanced encryption scheme, the wiretapper faces a problem of cryptanalysis after a communication channel with bits deletion and a legitimate party faces a problem of decryption after a channel with bit erasures. This paper proposes the encryption-decryption paradigm for the security enhancement of lightweight block ciphers based on dedicated error-correction coding and a simulator of the deletion channel controlled by the secret key. The security enhancement is analyzed in terms of the related probabilities, equivocation, mutual information and channel capacity. The cryptographic evaluation of the enhanced encryption includes employment of certain recent results regarding the upper-bounds on the capacity of channels with deletion errors. It is shown that the probability of correct classification which determines the cryptographic security depends on the deletion channel capacity, i.e., the equivocation after this channel, and number of codewords in employed error-correction coding scheme. Consequently, assuming that the basic encryption scheme has certain security level, it is shown that the security enhancement factor is a function of the deletion rate and dimension of the vectors subject to error-correction encoding, i.e., dimension of the encryption block.

Two-weight codes: Upper bounds and new optimal constructions

In this paper, we consider q-ary codes of length n and minimum Hamming distance d, which have two weights w1 and w2. These codes are denoted by (n,d,{w1,w2})q codes. Let Aq(n,d,{w1,w2}) denote the largest possible number of codewords in an (n,d,{w1,w2})q code and we simply write A(n,d,{w1,w2}) for A2(n,d,{w1,w2}). Some upper bounds on Aq(n,d,{w1,w2}) are given. The equivalences between binarytwo-weight codes and special combinatorial configurations with certain properties are established and then new upper bounds on A(n,d,{w1,w2}) are derived. For w1,w2∈{2,3,4}, optimal constructions of (n,d,{w1,w2})2 codes are presented. The exact value of A(n,d,{2,3}) is completely determined for all n and d. We determine the exact value of A(n,d,{2,4}) for any positive integer n≡2,4(mod6) and d∈{3,4}. The exact value of A(n,d,{3,4}) is determined for any integer n and d∈{6,7}, or d=5 and n≡0,2,3,11(mod12).

Detecting and Predicting Tipping Points

Tipping points are sudden, and sometimes irreversible and catastrophic, changes in a system’s dynamical regime. Complex networks are now widely used in the analysis of time series from a complex system. In this paper, we investigate the scope of network methods to indicate tipping points. In particular, we verify that the permutation entropy of transition networks constructed from time series observations of the logistic map can distinguish periodic and chaotic regimes and indicate bifurcations. The permutation entropy of transition networks, the mean edge betweenness of visibility graphs and the number of code words in compression networks, are each shown to indicate the onset of transition of a pitchfork bifurcation system. Our study shows that network methods are effective in detecting transitions. Network-based forecasts can be applied to models of real systems, as we illustrate by considering a lake eutrophication model.

RT distance and weight distributions of Type 1 constacyclic codes of length 4ps over Fpm[u

© Tubitak. For any odd prime p such that p m ≡ 1 (mod 4), the class of Λ-constacyclic codes of length 4p s over the finite commutative chain ring R a =F p m[u]/〈u a 〉 =F p m + uF p m + . . . + ua -1 F p m, for all units Λ of R a that have the form Λ = Λ 0 +uΛ 1 +. . .+u a-1 Λ a-1 , where Λ 0 ;Λ 1 ;: :: ;Λ a-1 ∈ F p m, Λ 0 ≠0; Λ 1 ≠0, is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a -λ-constacyclic code and a λ-constacyclic code of length 2p s . In the main case that the unit Λ is not a square, we show that any nonzero polynomial of degree < 4 over F p m is invertible in the ambient ring R a [x] /〈x 4ps -Λ〉 and use it to prove that the ambient ring 〈 a [x] /x 4ps -Λ〉 is a chain ring with maximal ideal 〈x 4 - λ 0 〉, where λ 0ps = λ 0 : As an application, the number of codewords and the dual of each λ-constacyclic code are provided. Furthermore, we get the Rosenbloom-Tsfasman (RT) distance and weight distributions of such codes. Using these results, the unique MDS code with respect to the RT distance is identified.

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].

A 3D shape measurement method based on novel segmented quantization phase coding

Codeword is very important in phase unwrapping based on phase coding method. Each phase coding fringe is marked with a unique codeword and then used to determine the fringe order for absolute phase retrieval. However, the traditional phase coding method is limited by the number of codeword. To overcome this problem, a segmented quantization phase coding and decoding method is proposed to achieve absolute phase retrieval without reducing the difference between adjacent quantized phase values or increasing the quantization levels. The quantized phase is modulated by a specific coding sequence and then embedded into phase coding fringe patterns, thereby greatly improving the accuracy of the decoding. The coding creates multiple segments of the whole region and each segment embeds the special coding sequence. Through the corresponding decoding algorithm, the fringe order can be obtained, and following this, the absolute phase is obtained. Simulation and experimental results demonstrate the feasibility and superiority of the proposed method to measure both a single object and multiple isolated objects.

Constructing self-dual cyclic codes over $${\mathbb {Z}}_{9}$$ Z 9 of length 3n

In this paper, we study cyclic codes over $$\mathbb {Z}_9$$ of length 3n, where n is a positive integer satisfying $$\mathrm{gcd}(3,n)=1$$ . First, a canonical form decomposition of any cyclic code over $$\mathbb {Z}_9$$ of length 3n are given and a unique set of generators for each subcode is presented. Hence the structure of any cyclic code over $$\mathbb {Z}_9$$ of length 3n is determined. From this decomposition, formulas for the number of all codes and the number of codewords in each code are given. Then dual codes and self-duality of these codes are investigated. As an application, all 10061824 distinct cyclic codes over $$\mathbb {Z}_9$$ of length 24 and all 544 self-dual codes among them are listed explicitly. Moreover, 280 new and good self-dual cyclic codes over $$\mathbb {Z}_9$$ with basic parameters $$\left( 24, 3^{24}, 3\right) $$ are obtained.

Improved bounds on 2-frameproof codes with length 4

Frameproof codes (FPCs) are widely studied as they are classic fingerprinting codes that can protect copyrighted materials. The main interests are construction methods and bounds of the number of codewords of FPCs for a fixed length when the alphabet size approaches infinity. In this paper, we focus on the upper bound of the size of FPCs when the fixed length is 4 and the strength is 2. We obtain an upper bound $$2q^2-2q+7$$ on the size of a q-ary 2-FPC of length 4 for any positive integer $$q> 48$$ . The best previously well known bound of this type of FPCs is $$2q^2-2$$ , which is due to Blackburn (SIAM J Discret Math 16:499–510, 2003). Our new upper bound improves the previous upper bound and it is not very far from the current best lower bound $$2q^2-4q+3$$ obtained from the explicit construction due to Chee and Zhang (IEEE Trans Inf Theory 58:5449–5453, 2012).

Negacyclic codes of length [formula omitted] over [formula omitted] and their duals

For any odd prime p, negacyclic codes of length 4ps over the finite commutative chain ring Fpm+uFpm are investigated. The algebraic structures of such codes are classified and completely determined. As an application, the number of codewords and the dual of each negacyclic code are obtained. Simpler structure of cyclic codes of length 4ps over Fpm+uFpm is also noted. Among others, some self-dual negacyclic and cyclic codes of length 4ps over Fpm+uFpm are provided.

Constructions for optimal cyclic ternary constant-weight codes of weight four and distance six

A cyclic (n,d,w)q code is a q-ary cyclic code of length n, minimum Hamming distance d and weight w. In this paper, we investigate cyclic (n,6,4)3 codes. A new upper bound on CA3(n,6,4), the largest possible number of codewords in a cyclic (n,6,4)3 code, is given. Two new constructions for optimal cyclic (n,6,4)3 codes based on cyclic (n,4,1) difference packings are presented. As a consequence, the exact value of CA3(n,6,4) is determined for any positive integer n≡0,6,18(mod24). We also obtain some other infinite classes of optimal cyclic (n,6,4)3 codes.

All α + uβ-constacyclic codes of length nps over [formula omitted

Let Fpm be a finite field with cardinality pm and R=Fpm+uFpm with u2=0. We aim to determine all α+uβ-constacyclic codes of length nps over R, where α,β∈Fpm⁎, n,s∈N+ and gcd⁡(n,p)=1. Let α0∈Fpm⁎ and α0ps=α. The residue ring R[x]/〈xnps−α−uβ〉 is a chain ring with the maximal ideal 〈xn−α0〉 in the case that xn−α0 is irreducible in Fpm[x]. If xn−α0 is reducible in Fpm[x], we give the explicit expressions of the ideals of R[x]/〈xnps−α−uβ〉. Besides, the number of codewords and the dual code of every α+uβ-constacyclic code are provided.