Journal articles: 'Linear Subspace Codes'

1

Poroch, Mahdieh Hakimi, and Ali Asghar Talebi. "Product of symplectic groups and its cyclic orbit code." Discrete Mathematics, Algorithms and Applications 11, no. 05 (October 2019): 1950061. http://dx.doi.org/10.1142/s1793830919500617.

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Constant dimension subspace codes are subsets of the finite Grassmann Variety. Orbit codes are constant dimension subspace codes that arise as the orbit of subgroup of general linear group acting on subspaces in an ambient space. In particular, orbit codes of symplectic subgroup of the general linear group have been investigated recently. In this paper, we determine product of symplectic groups and its orbit code, and decoding algorithm of this code is considered.

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Gao, You, and Gang Wang. "Bounds on Subspace Codes Based on Subspaces of Type(m,1)in Singular Linear Space." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/497958.

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The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codesn+l,M,d,(m,1)qbased on subspaces of type(m,1)in singular linear spaceFq(n+l)over finite fieldsFqare presented. Then, we prove that codes based on subspaces of type(m,1)in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures inFq(n+l).

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Zanardi, Paolo, and Mario Rasetti. "Error Avoiding Quantum Codes." Modern Physics Letters B 11, no. 25 (October 30, 1997): 1085–93. http://dx.doi.org/10.1142/s0217984997001304.

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The existence is proved of a class of open quantum systems that admits a linear subspace [Formula: see text] of the space of states such that the restriction of the dynamical semigroup to the states built over [Formula: see text] is unitary. Such subspace allows for error-avoiding (noiseless) encoding of quantum information.

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Li, Xia, Qin Yue, and Deng Tang. "A family of linear codes from constant dimension subspace codes." Designs, Codes and Cryptography 90, no. 1 (October 27, 2021): 1–15. http://dx.doi.org/10.1007/s10623-021-00960-x.

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Gorla, Elisa, and Alberto Ravagnani. "Subspace codes from Ferrers diagrams." Journal of Algebra and Its Applications 16, no. 07 (July 7, 2016): 1750131. http://dx.doi.org/10.1142/s0219498817501316.

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In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

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Wang, Gang, Min-Yao Niu, and Fang-Wei Fu. "Bounds on Subspace Codes Based on Orthogonal Space Over Finite Fields of Characteristic 2." International Journal of Foundations of Computer Science 30, no. 05 (August 2019): 735–57. http://dx.doi.org/10.1142/s0129054119500199.

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In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over finite fields [Formula: see text] of characteristic 2, denoted by [Formula: see text], are presented, where [Formula: see text] is a positive integer, [Formula: see text], [Formula: see text], [Formula: see text]. Then, we prove that [Formula: see text] codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text], where [Formula: see text] denotes the collection of all the [Formula: see text]-dimensional totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code [Formula: see text] in [Formula: see text] are provided, where [Formula: see text] denotes the collection of all the totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2.

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Zullo, Ferdinando. "Multi-orbit cyclic subspace codes and linear sets." Finite Fields and Their Applications 87 (March 2023): 102153. http://dx.doi.org/10.1016/j.ffa.2022.102153.

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Guruswami, Venkatesan, and Chaoping Xing. "Optimal Rate List Decoding over Bounded Alphabets Using Algebraic-geometric Codes." Journal of the ACM 69, no. 2 (April 30, 2022): 1–48. http://dx.doi.org/10.1145/3506668.

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We give new constructions of two classes of algebraic code families that are efficiently list decodable with small output list size from a fraction 1-R-ε of adversarial errors, where R is the rate of the code, for any desired positive constant ε. The alphabet size depends only ε and is nearly optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield . In both cases, we develop a linear-algebraic approach to perform list decoding, which pins down the candidate messages to a subspace with a nice “periodic” structure. To prune this subspace and obtain a good bound on the list size, we pick subcodes of these codes by pre-coding into certain subspace-evasive sets that are guaranteed to have small intersection with the sort of periodic subspaces that arise in our list decoding. We develop two approaches for constructing such subspace-evasive sets. The first is a Monte Carlo construction of hierearchical subspace-evasive (h.s.e.) sets that leads to excellent list size but is not explicit. The second approach exploits a further ultra-periodicity of our subspaces and uses a novel construct called subspace designs , which were subsequently constructed explicitly and also found further applications in pseudorandomness. To get a family of codes over a fixed alphabet size, we instantiate our approach with algebraic-geometric codes based on the Garcia–Stichtenoth tower of function fields. Combining this with pruning via h.s.e. sets yields codes list-decodable up to a 1-R-ε error fraction with list size bounded by O (1/ε), matching the existential bound for random codes up to constant factors. Further, the alphabet size can be made exp ( Õ (1/ε 2 )), which is not much worse than the lower bound of exp (Ω (1/ε)). The parameters we achieve are thus quite close to the existential bounds in all three aspects (error-correction radius, alphabet size, and list size) simultaneously. This construction is, however, Monte Carlo and the claimed list-decoding property only holds with high probability. Once the code is (efficiently) sampled, the encoding/decoding algorithms are deterministic with a running time O _ε ( N c ) for an absolute constant c , where N is the code’s block length. Using subspace designs instead for the pruning, our approach yields the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-ε fraction of errors over an alphabet of constant size exp (Õ(1/ε 2 )). The list-size bound is upper bounded by a very slowly growing function of the block length N ; in particular, it is at most O(log ( r ) N ) (the r th iterated logarithm) for any fixed integer r . The explicit construction avoids the shortcoming of the Monte Carlo sampling at the expense of a slightly worse list size.

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Li, Guanyue, Qianfen Jiao, Sheng Qian, Si Wu, and Hau-San Wong. "High Fidelity GAN Inversion via Prior Multi-Subspace Feature Composition." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 8366–74. http://dx.doi.org/10.1609/aaai.v35i9.17017.

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Generative Adversarial Networks (GANs) have shown impressive gains in image synthesis. GAN inversion was recently studied to understand and utilize the knowledge it learns, where a real image is inverted back to a latent code and can thus be reconstructed by the generator. Although increasing the number of latent codes can improve inversion quality to a certain extent, we find that important details may still be neglected when performing feature composition over all the intermediate feature channels. To address this issue, we propose a Prior multi-Subspace Feature Composition (PmSFC) approach for high-fidelity inversion. Considering that the intermediate features are highly correlated with each other, we incorporate a self-expressive layer in the generator to discover meaningful subspaces. In this case, the features at a channel can be expressed as a linear combination of those at other channels in the same subspace. We perform feature composition separately in the subspaces. The semantic differences between them benefit the inversion quality, since the inversion process is regularized based on different aspects of semantics. In the experiments, the superior performance of PmSFC demonstrates the effectiveness of prior subspaces in facilitating GAN inversion together with extended applications in visual manipulation.

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F. N. Al-Jobory, Jinan, Emad B. Al-Zangana, and Faez Hassan Ali. "Modular Irreducible Representations of the FpW4-Submodules ,()pFNof the Modules ,()pFMas Linear Codes, where W4is the Weyl Group of Type B4." Al-Nahrain Journal of Science 24, no. 2 (June 1, 2021): 48–63. http://dx.doi.org/10.22401/anjs.24.2.08.

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The modular representations of the FpWn-Specht modules( , )KSas linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules( , )pFNof the Specht modules pFS ( , )as linear codes where W4is the Weyl group of type B4is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules( , )pFNof the FpW4-modules( , )pFMfor each pair of partitions( , )of a positive integer n4, where FpGF(p) is the Galois field (finite field) of order p, and pis a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace((2,1),(1))()pU representing the irreducible FpW4-submodules((2,1),(1))pFNof the FpW4-modules((2,1),(1))pF Mand give the linear code of ((2,1),(1))()pU for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces( , )()pUfor all pair of partitions( , )of a positive integer n4, and for each prime number p greater than or equal to 3.We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn

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Etzion, Tuvi, and Antonia Wachter-Zeh. "Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding." IEEE Transactions on Information Theory 64, no. 4 (April 2018): 2460–73. http://dx.doi.org/10.1109/tit.2018.2797183.

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Gadouleau, Maximilien, and Zhiyuan Yan. "Packing and Covering Properties of Subspace Codes for Error Control in Random Linear Network Coding." IEEE Transactions on Information Theory 56, no. 5 (May 2010): 2097–108. http://dx.doi.org/10.1109/tit.2010.2043780.

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13

Nguyen, Hai, Jonathan Wittmer, and Tan Bui-Thanh. "DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems." Computation 10, no. 3 (March 11, 2022): 38. http://dx.doi.org/10.3390/computation10030038.

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This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

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Yaremchuk, Max, Dmitri Nechaev, and Gleb Panteleev. "A Method of Successive Corrections of the Control Subspace in the Reduced-Order Variational Data Assimilation*." Monthly Weather Review 137, no. 9 (September 1, 2009): 2966–78. http://dx.doi.org/10.1175/2009mwr2592.1.

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Abstract A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

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Mo, Tieqiang, and Renfa Li. "Iteratively solving sparse linear system based on PaRSEC task scheduling." International Journal of High Performance Computing Applications 34, no. 3 (January 13, 2020): 306–15. http://dx.doi.org/10.1177/1094342019899997.

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With the new architecture and new programming paradigms such as task-based scheduling emerging in the parallel high performance computing area, it is of great importance to utilize these features to tune the monolithic computing codes. In this article, the classical conjugate gradient algorithms targeting at sparse linear system Ax = b in Krylov subspace are pipelining to execute interdependent tasks on Parallel Runtime Scheduling and Execution Controller (PaRSEC) runtime. Firstly, the sparse matrix A is split in rows to unfold more coarse-grained parallelism. Secondly, the partitioned sub-vectors are not assembled into one full vector in RAM to run sparse matrix–vector product (SpMV) operations for eliminating the communication overhead. Moreover, in the SpMV computation, if all elements of one column in the split sub-matrix are zeros, the corresponding product operations of these elements may be removed by reorganizing sub-vectors. Finally, the latency of migrating sub-vector is partially overlapped by the duration of performing SpMV operations through the further splitting in columns of sparse matrix on GPUs. In experiments, a series of tests demonstrate that optimal speedup and higher pipelining efficiency has been achieved for the pipelined task scheduling on PaRSEC runtime. Fusing SpMV concurrency and dot product pipelining can achieve higher speedup and efficiency.

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Zhu, Jianchen, Shengjie Zhao, and Di Wu. "Classification of Remote Sensing Images Through Reweighted Sparse Subspace Representation Using Compressed Data." Traitement du Signal 38, no. 1 (February 28, 2021): 27–37. http://dx.doi.org/10.18280/ts.380103.

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In many real-world scenarios, subspace clustering essentially aims to cluster unlabeled high-dimensional data into a union of finite-dimensional linear subspaces. The problem is that the data are always high-dimensional, with the increase of the computation, storge, and communication of various intelligent data-driven systems. This paper attempts to develop a method to cluster spectral images directly using the measurements of compressive coded aperture snapshot spectral imager (CASSI), eliminating the need to reconstruct the entire data cube. Assuming that compressed measurements are drawn from multiple subspaces, a novel algorithm was developed by solving a 1-norm minimization problem, which is known as reweighted sparse subspace clustering (RSSC). The proposed algorithm clusters the compressed measurements into different subspaces, which greatly improves the clustering accuracy over the SSC algorithm by adding a reweighted step. The compressed CASSI measurements obtained using the coherence-based coded aperture can improve the performance of the proposed spectral image clustering method. The accuracy of our spectral image clustering approach was verified through simulations on two real datasets.

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García-Planas, María Isabel, Maria Dolors Magret, and Laurence Emilie Um. "Monomial codes seen as invariant subspaces." Open Mathematics 15, no. 1 (August 23, 2017): 1099–107. http://dx.doi.org/10.1515/math-2017-0093.

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Abstract It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

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Sosa-Gómez, Guillermo, Octavio Paez-Osuna, Omar Rojas, Pedro Luis del Ángel Rodríguez, Herbert Kanarek, and Evaristo José Madarro-Capó. "Construction of Boolean Functions from Hermitian Codes." Mathematics 10, no. 6 (March 11, 2022): 899. http://dx.doi.org/10.3390/math10060899.

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In 2005, Guillot published a method for the construction of Boolean functions using linear codes through the Maiorana–McFarland construction of Boolean functions. In this work, we present a construction using Hermitian codes, starting from the classic Maiorana–McFarland construction. This new construction describes how the set of variables is divided into two complementary subspaces, one of these subspaces being a Hermitian Code. The ideal theoretical parameters of the Hermitian code are proposed to reach desirable values of the cryptographic properties of the constructed Boolean functions such as nonlinearity, resiliency order, and order of propagation. An extension of Guillot’s work is also made regarding parameters selection using algebraic geometric tools, including explicit examples.

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Klappenecker, Andreas, and Pradeep Kiran Sarvepalli. "On subsystem codes beating the quantum Hamming or Singleton bound." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2087 (August 21, 2007): 2887–905. http://dx.doi.org/10.1098/rspa.2007.0028.

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Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

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Gruica, Anina, and Alberto Ravagnani. "Common Complements of Linear Subspaces and the Sparseness of MRD Codes." SIAM Journal on Applied Algebra and Geometry 6, no. 2 (April 11, 2022): 79–110. http://dx.doi.org/10.1137/21m1428947.

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Wang, Congcong, Yingying Zhang, Zhuoqun Li, Xiaona Zhang, and You Gao. "The construction of LDPC codes based on the subspaces of singular linear space over finite field." Discrete Mathematics, Algorithms and Applications 08, no. 04 (November 8, 2016): 1650073. http://dx.doi.org/10.1142/s1793830916500737.

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Let [Formula: see text] be a finite field with [Formula: see text] elements, where [Formula: see text] is a prime power. [Formula: see text] denotes the [Formula: see text]-dimensional row linear space over [Formula: see text]. In this paper, we construct a series of LDPC codes based on the subspaces of singular linear space over [Formula: see text], and calculate their parameters.

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Alonso-González, Clementa, and Miguel Ángel Navarro-Pérez. "On Generalized Galois Cyclic Orbit Flag Codes." Mathematics 10, no. 2 (January 11, 2022): 217. http://dx.doi.org/10.3390/math10020217.

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Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.

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Wang, Gang, Min-Yao Niu, and Fang-Wei Fu. "Constructions of (r,t)-LRC Based on Totally Isotropic Subspaces in Symplectic Space Over Finite Fields." International Journal of Foundations of Computer Science 31, no. 03 (April 2020): 327–39. http://dx.doi.org/10.1142/s0129054120500112.

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Linear code with locality [Formula: see text] and availability [Formula: see text] is that the value at each coordinate [Formula: see text] can be recovered from [Formula: see text] disjoint repairable sets each containing at most [Formula: see text] other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of [Formula: see text]-locally repairable linear codes based on totally isotropic subspaces in symplectic space [Formula: see text] over finite fields [Formula: see text] are presented. Meanwhile, comparisons are made among the [Formula: see text]-locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate [Formula: see text] and relative distance [Formula: see text].

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JAMIOŁKOWSKI, ANDRZEJ. "ON APPLICATIONS OF PI-ALGEBRAS IN THE ANALYSIS OF QUANTUM CHANNELS." International Journal of Quantum Information 10, no. 08 (December 2012): 1241007. http://dx.doi.org/10.1142/s0219749912410079.

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In this paper, we discuss some constructive procedures which can be used in characterizations of linear transformations which preserve the set of states of a fixed quantum system. Our methods are based on analyzing an explicit form of a linear positive map in its Kraus representation. In particular, we discuss the so-called partial commutativity of operators and its applications to investigation of decoherence-free subspaces. These subspaces can also be considered as a special class of quantum error correcting codes. Using the concept of standard polynomials and Amitsur–Levitzki theorem and other ideas from the so-called polynomial identity algebras (PI-algebras) we discuss some effective algorithms for analyzing properties of quantum channels.

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Ghorpade, Sudhir R., Arunkumar R. Patil, and Harish K. Pillai. "Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes." Finite Fields and Their Applications 15, no. 1 (February 2009): 54–68. http://dx.doi.org/10.1016/j.ffa.2008.08.001.

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Rakdi, M. A., and N. Midoune. "Weights of the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$ over a finite field." Carpathian Mathematical Publications 11, no. 2 (December 31, 2019): 422–30. http://dx.doi.org/10.15330/cmp.11.2.422-430.

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Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.

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Ganian, Robert, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. "On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 3906–13. http://dx.doi.org/10.1609/aaai.v34i04.5804.

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We consider a fundamental matrix completion problem where we are given an incomplete matrix and a set of constraints modeled as a CSP instance. The goal is to complete the matrix subject to the input constraints and in such a way that the complete matrix can be clustered into few subspaces with low rank. This problem generalizes several problems in data mining and machine learning, including the problem of completing a matrix into one with minimum rank. In addition to its ubiquitous applications in machine learning, the problem has strong connections to information theory, related to binary linear codes, and variants of it have been extensively studied from that perspective. We formalize the problem mentioned above and study its classical and parameterized complexity. We draw a detailed landscape of the complexity and parameterized complexity of the problem with respect to several natural parameters that are desirably small and with respect to several well-studied CSP fragments.

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Xie, Xuping, Guannan Zhang, and Clayton G. Webster. "Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network." Mathematics 7, no. 8 (August 19, 2019): 757. http://dx.doi.org/10.3390/math7080757.

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In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams–Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches.

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Mitskopoulos, Lazaros, and Arno Onken. "Discovering Low-Dimensional Descriptions of Multineuronal Dependencies." Entropy 25, no. 7 (July 6, 2023): 1026. http://dx.doi.org/10.3390/e25071026.

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Coordinated activity in neural populations is crucial for information processing. Shedding light on the multivariate dependencies that shape multineuronal responses is important to understand neural codes. However, existing approaches based on pairwise linear correlations are inadequate at capturing complicated interaction patterns and miss features that shape aspects of the population function. Copula-based approaches address these shortcomings by extracting the dependence structures in the joint probability distribution of population responses. In this study, we aimed to dissect neural dependencies with a C-Vine copula approach coupled with normalizing flows for estimating copula densities. While this approach allows for more flexibility compared to fitting parametric copulas, drawing insights on the significance of these dependencies from large sets of copula densities is challenging. To alleviate this challenge, we used a weighted non-negative matrix factorization procedure to leverage shared latent features in neural population dependencies. We validated the method on simulated data and applied it on copulas we extracted from recordings of neurons in the mouse visual cortex as well as in the macaque motor cortex. Our findings reveal that neural dependencies occupy low-dimensional subspaces, but distinct modules are synergistically combined to give rise to diverse interaction patterns that may serve the population function.

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Honold, Thomas, Michael Kiermaier, and Sascha Kurz. "Johnson Type Bounds for Mixed Dimension Subspace Codes." Electronic Journal of Combinatorics 26, no. 3 (August 30, 2019). http://dx.doi.org/10.37236/8188.

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Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

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31

Kumar, Ashwini, and R. S. Raja Durai. "Construction of LCD-MRD codes of length n > N." Discrete Mathematics, Algorithms and Applications, April 13, 2021, 2150117. http://dx.doi.org/10.1142/s1793830921501172.

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An MRD code [Formula: see text] is a [Formula: see text]-dimensional [Formula: see text]-linear subspace of the [Formula: see text]-dimensional vector space [Formula: see text] over [Formula: see text] for [Formula: see text]. Linear codes [Formula: see text] (of length [Formula: see text], dimension [Formula: see text]) that are obtained from MRD codes [Formula: see text] satisfying the property [Formula: see text] are defined to be LCD MRD codes. An inherent relationship between the generator G and parity-check matrices H of LCD MRD codes is observed. This in fact identifies LCD MRD codes as trivial ([Formula: see text]) and nontrivial ([Formula: see text]) codes. Further, two classes of LCD MRD codes of length [Formula: see text] are constructed from [Formula: see text] (trivial and nontrivial) LCD MRD codes of length [Formula: see text]. Examples are provided for demonstration of results obtained.

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Fourier, Ghislain, and Gabriele Nebe. "Degenerate flag varieties in network coding." Advances in Mathematics of Communications, 2021, 0. http://dx.doi.org/10.3934/amc.2021027.

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Building upon the application of flags to network coding introduced in [<xref ref-type="bibr" rid="b6">6</xref>], we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding.

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Gluesing-Luerssen, Heide, and Hunter Lehmann. "Automorphism groups and isometries for cyclic orbit codes." Advances in Mathematics of Communications, 2021, 0. http://dx.doi.org/10.3934/amc.2021040.

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We study orbit codes in the field extension <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q^n} $\end{document}</tex-math></inline-formula>. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_{q^n} $\end{document}</tex-math></inline-formula>. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

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Kramer, Joshua Brown. "On the most Weight $w$ Vectors in a Dimension $k$ Binary Code." Electronic Journal of Combinatorics 17, no. 1 (October 29, 2010). http://dx.doi.org/10.37236/414.

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Ahlswede, Aydinian, and Khachatrian posed the following problem: what is the maximum number of Hamming weight $w$ vectors in a $k$-dimensional subspace of $\mathbb{F}_2^n$? The answer to this question could be relevant to coding theory, since it sheds light on the weight distributions of binary linear codes. We give some partial results. We also provide a conjecture for the complete solution when $w$ is odd as well as for the case $k \geq 2w$ and $w$ even. One tool used to study this problem is a linear map that decreases the weight of nonzero vectors by a constant. We characterize such maps.

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Schauz, Uwe. "Anti-Codes in Terms of Berlekamp's Switching Game." Electronic Journal of Combinatorics 19, no. 1 (January 6, 2012). http://dx.doi.org/10.37236/17.

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We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the $n$-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$. Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.

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36

Kurz, Sascha. "The interplay of different metrics for the construction of constant dimension codes." Advances in Mathematics of Communications, 2022, 0. http://dx.doi.org/10.3934/amc.2021069.

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A basic problem for constant dimension codes is to determine the maximum possible size <inline-formula><tex-math id="M1">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> of a set of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-dimensional subspaces in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_q^n $\end{document}</tex-math></inline-formula>, called codewords, such that the subspace distance satisfies <inline-formula><tex-math id="M4">\begin{document}$ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $\end{document}</tex-math></inline-formula> for all pairs of different codewords <inline-formula><tex-math id="M5">\begin{document}$ U $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for <inline-formula><tex-math id="M7">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show the potential for further improvements. As examples we give improved constructions for the cases <inline-formula><tex-math id="M8">\begin{document}$ A_q(10,4;5) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ A_q(11,4;4) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ A_q(12,6;6) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ A_q(15,4;4) $\end{document}</tex-math></inline-formula>. We also derive general upper bounds for subcodes arising in those constructions.

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Mogilnykh, Ivan. "Completely Regular Codes in Johnson and Grassmann Graphs with Small Covering Radii." Electronic Journal of Combinatorics 29, no. 2 (June 17, 2022). http://dx.doi.org/10.37236/10083.

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Let ${\cal L}$ be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the $4$-subspaces, which do not contain subspaces from ${\cal L}$ is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely regular code in the Johnson graph $J(n,6)$ from the Steiner quadruple system of the extended Hamming code. We obtain several new completely regular codes with covering radius $1$ in the Grassmann graph $J_2(6,3)$ using binary linear programming.

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Pelen, Rumi Melih. "Three weight ternary linear codes from non-weakly regular bent functions." Advances in Mathematics of Communications, 2022, 0. http://dx.doi.org/10.3934/amc.2022020.

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This paper constructs several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions based on a generic construction method. Instead of the whole space, we use the subspaces <inline-formula><tex-math id="M1">\begin{document}$ B_{\pm}(f) $\end{document}</tex-math></inline-formula> associated with a ternary non-weakly regular dual-bent function <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Unusually, we use the pre-image sets of the dual function <inline-formula><tex-math id="M3">\begin{document}$ f^* $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ B_{\pm}(f) $\end{document}</tex-math></inline-formula> as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dimension. We represent the weight distribution of the constructed codes, and we also give several examples.

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Hekrdla, M. "A Constellation Space Dimensionality Reduced Sub-Optimal Receiver for Orthogonal STBC CPM Modulation in a MIMO Channel." Acta Polytechnica 49, no. 2 (January 2, 2009). http://dx.doi.org/10.14311/1107.

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We consider burst orthogonal space-time block coded (OSTBC) CPM modulation in a MIMO flat slow Rayleigh fading channel. The optimal receiver must process a multidimensional non-linear CPM signal on each antenna. This task imposes a high load on the receiver computational performance and increases its complexity. We analytically derive a suboptimal receiver with a reduced number of front end matched filters (MFs) corresponding to the CPM dimension. Our derivation is made fully in the constellation signal space, and the reduction is based on the linear orthogonal projection to the optimal subspace. Criterion optimality is a standard space-time rank and determinant criterion. The optimal arbitrary-dimensional subspace search leads to the eigenvector solution. We present the condition on a sufficient subspace dimension and interpret the meaning of the corresponding eigenvalues. It is shown that the determinant and rank criterion for OSTBC CPM is equivalent to the uncoded CPM Euclidean distance criterion. Hence the proposed receiver may be practical for uncoded CPM and foremost in a serially concatenated (SC) CPM system. All the derivations are supported by suitable error simulations for binary 2REC h= 1/2, but the procedure is generally valid for any CPM variant. We consider OSTBC CPM in a Rayleigh fading AWGN channel and SC CPM in an AWGN channel.

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40

Fine, Justin M., David J.-N. Maisson, Seng Bum Michael Yoo, Tyler V. Cash-Padgett, Maya Zhe Wang, Jan Zimmermann, and Benjamin Y. Hayden. "Abstract value encoding in neural populations but not single neurons." Journal of Neuroscience, May 19, 2023, JN—RM—1954–22. http://dx.doi.org/10.1523/jneurosci.1954-22.2023.

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An important open question in neuroeconomics is how the brain represents the value of offers in a way that is both abstract (allowing for comparison) and concrete (preserving the details of the factors that influence value). Here we examine neuronal responses to risky and safe options in five brain regions that putatively encode value in male macaques. Surprisingly, we find no detectable overlap in the neural codes used for risky and safe options, even when the options have identical subjective values (as revealed by preference) in any of the regions. Indeed, responses are not just uncorrelated but occupy distinct (semi-orthogonal) encoding subspaces. Notably, however, these subspaces are linked through a linear transform of their constituent encodings, a property that allows for comparison of dissimilar option types. This encoding scheme allows these regions to have their cake and eat it too: they can encode the detailed factors that influence offer value (here, risky and safety) but also directly compare dissimilar offer types. Together these results suggest a neuronal basis for the qualitatively different psychological properties of risky and safe options and highlight the power of population geometry to resolve outstanding problems in neural coding.SIGNIFICANCE STATEMENT:To make economic choices, we must have some mechanism for comparing dissimilar offers. We propose that the brain uses distinct neural codes for risky and safe offers, but that these codes are linearly transformable. This encoding scheme has the twin advantages of allowing for comparison across offer types while preserving information about offer type, which in turn allows for flexibility in changing circumstances. We show that responses to risky and safe offers exhibit these predicted properties in five different reward-sensitive regions. Together, these results highlight the power of population coding principles for solving representation problems in economic choice.

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Haas, Wolfgang. "On the Failing Cases of the Johnson Bound for Error-Correcting Codes." Electronic Journal of Combinatorics 15, no. 1 (April 18, 2008). http://dx.doi.org/10.37236/779.

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A central problem in coding theory is to determine $A_q(n,2e+1)$, the maximal cardinality of a $q$-ary code of length $n$ correcting up to $e$ errors. When $e$ is fixed and $n$ is large, the best upper bound for $A(n,2e+1)$ (the binary case) is the well-known Johnson bound from 1962. This however simply reduces to the sphere-packing bound if a Steiner system $S(e+1,2e+1,n)$ exists. Despite the fact that no such system is known whenever $e\geq 5$, they possibly exist for a set of values for $n$ with positive density. Therefore in these cases no non-trivial numerical upper bounds for $A(n,2e+1)$ are known. In this paper the author demonstrates a technique for upper-bounding $A_q(n,2e+1)$, which closes this gap in coding theory. The author extends his earlier work on the system of linear inequalities satisfied by the number of elements of certain codes lying in $k$-dimensional subspaces of the Hamming Space. The method suffices to give the first proof, that the difference between the sphere-packing bound and $A_q(n,2e+1)$ approaches infinity with increasing $n$ whenever $q$ and $e\geq 2$ are fixed. A similar result holds for $K_q(n,R)$, the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Moreover the author presents a new bound for $A(n,3)$ giving for instance $A(19,3)\leq 26168$.

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Kar, Arjun. "Non-isometric quantum error correction in gravity." Journal of High Energy Physics 2023, no. 2 (February 20, 2023). http://dx.doi.org/10.1007/jhep02(2023)195.

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Abstract We construct and study an ensemble of non-isometric error correcting codes in a toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear transformation with independent complex Gaussian random entries of zero mean and unit variance. Using measure concentration, we show that the typical such code is very likely to preserve pairwise inner products in a set S of states that can be subexponentially large in the microcanonical Hilbert space dimension of the black hole. The size of this set also serves as an upper limit on the bulk effective field theory Hilbert space dimension. Similar techniques are used to demonstrate the existence of state-specific reconstructions of S-preserving code space unitary operators. State-specific reconstructions on subspaces exist when they are expected to by entanglement wedge reconstruction. We comment on relations to complexity theory and the breakdown of bulk effective field theory.

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Journal articles on the topic 'Linear Subspace Codes'

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