Recent advancements in learned image compression (LIC) methods have demonstrated superior performance over traditional hand-crafted codecs. These learning-based methods often employ convolutional neural networks (CNNs) or Transformer-based architectures. However, these nonlinear approaches frequently overlook the frequency characteristics of images, which limits their compression efficiency. To address this issue, we propose a novel Transformer-based image compression method that enhances the transformation stage by considering frequency components within the feature map. Our method integrates a novel Hybrid Spatial-Channel Attention Transformer Block (HSCATB), where a spatial-based branch independently handles high and low frequencies at the attention layer, and a Channel-aware Self-Attention (CaSA) module captures information across channels, significantly improving compression performance. Additionally, we introduce a Mixed Local-Global Feed Forward Network (MLGFFN) within the Transformer block to enhance the extraction of diverse and rich information, which is crucial for effective compression. These innovations collectively improve the transformation's ability to project data into a more decorrelated latent space, thereby boosting overall compression efficiency. Experimental results demonstrate that our framework surpasses state-of-the-art LIC methods in rate-distortion performance.
{"title":"Bi-Level Spatial and Channel-aware Transformer for Learned Image Compression","authors":"Hamidreza Soltani, Erfan Ghasemi","doi":"arxiv-2408.03842","DOIUrl":"https://doi.org/arxiv-2408.03842","url":null,"abstract":"Recent advancements in learned image compression (LIC) methods have\u0000demonstrated superior performance over traditional hand-crafted codecs. These\u0000learning-based methods often employ convolutional neural networks (CNNs) or\u0000Transformer-based architectures. However, these nonlinear approaches frequently\u0000overlook the frequency characteristics of images, which limits their\u0000compression efficiency. To address this issue, we propose a novel\u0000Transformer-based image compression method that enhances the transformation\u0000stage by considering frequency components within the feature map. Our method\u0000integrates a novel Hybrid Spatial-Channel Attention Transformer Block (HSCATB),\u0000where a spatial-based branch independently handles high and low frequencies at\u0000the attention layer, and a Channel-aware Self-Attention (CaSA) module captures\u0000information across channels, significantly improving compression performance.\u0000Additionally, we introduce a Mixed Local-Global Feed Forward Network (MLGFFN)\u0000within the Transformer block to enhance the extraction of diverse and rich\u0000information, which is crucial for effective compression. These innovations\u0000collectively improve the transformation's ability to project data into a more\u0000decorrelated latent space, thereby boosting overall compression efficiency.\u0000Experimental results demonstrate that our framework surpasses state-of-the-art\u0000LIC methods in rate-distortion performance.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Diego Ruano, Rodrigo San-José, Ivan Soprunov
In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a consequence, we show that a binary triorthogonal matrix uniquely determines the parameters of the corresponding triorthogonal quantum code, meaning that any other equivalent matrix that is also triorthogonal gives rise to a triorthogonal quantum code with the same parameters.
{"title":"Binary Triorthogonal and CSS-T Codes for Quantum Error Correction","authors":"Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Diego Ruano, Rodrigo San-José, Ivan Soprunov","doi":"arxiv-2408.02916","DOIUrl":"https://doi.org/arxiv-2408.02916","url":null,"abstract":"In this paper, we study binary triorthogonal codes and their relation to\u0000CSS-T quantum codes. We characterize the binary triorthogonal codes that are\u0000minimal or maximal with respect to the CSS-T poset, and we also study how to\u0000derive new triorthogonal matrices from existing ones. Given a binary\u0000triorthogonal matrix, we characterize which of its equivalent matrices are also\u0000triorthogonal. As a consequence, we show that a binary triorthogonal matrix\u0000uniquely determines the parameters of the corresponding triorthogonal quantum\u0000code, meaning that any other equivalent matrix that is also triorthogonal gives\u0000rise to a triorthogonal quantum code with the same parameters.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jaehyup Seong, Juha Park, Dong-Hyun Jung, Jeonghun Park, Wonjae Shin
Low Earth orbit (LEO) satellite communications (SATCOM) with ubiquitous global connectivity is deemed a pivotal catalyst in advancing wireless communication systems for 5G and beyond. LEO SATCOM excels in delivering versatile information services across expansive areas, facilitating both unicast and multicast transmissions via high-speed broadband capability. Nonetheless, given the broadband coverage of LEO SATCOM, traffic demand distribution within the service area is non-uniform, and the time/frequency/power resources available at LEO satellites remain significantly limited. Motivated by these challenges, we propose a rate-matching framework for non-orthogonal unicast and multicast (NOUM) transmission. Our approach aims to minimize the difference between offered rates and traffic demands for both unicast and multicast messages. By multiplexing unicast and multicast transmissions over the same radio resource, rate-splitting multiple access (RSMA) is employed to manage interference between unicast and multicast streams, as well as inter-user interference under imperfect channel state information at the LEO satellite. To address the formulated problems non-smoothness and non-convexity, the common rate is approximated using the LogSumExp technique. Thereafter, we represent the common rate portion as the ratio of the approximated function, converting the problem into an unconstrained form. A generalized power iteration (GPI)-based algorithm, coined GPI-RS-NOUM, is proposed upon this reformulation. Through comprehensive numerical analysis across diverse simulation setups, we demonstrate that the proposed framework outperforms various benchmarks for LEO SATCOM with uneven traffic demands.
{"title":"Rate-Splitting for Joint Unicast and Multicast Transmission in LEO Satellite Networks with Non-Uniform Traffic Demand","authors":"Jaehyup Seong, Juha Park, Dong-Hyun Jung, Jeonghun Park, Wonjae Shin","doi":"arxiv-2408.02872","DOIUrl":"https://doi.org/arxiv-2408.02872","url":null,"abstract":"Low Earth orbit (LEO) satellite communications (SATCOM) with ubiquitous\u0000global connectivity is deemed a pivotal catalyst in advancing wireless\u0000communication systems for 5G and beyond. LEO SATCOM excels in delivering\u0000versatile information services across expansive areas, facilitating both\u0000unicast and multicast transmissions via high-speed broadband capability.\u0000Nonetheless, given the broadband coverage of LEO SATCOM, traffic demand\u0000distribution within the service area is non-uniform, and the\u0000time/frequency/power resources available at LEO satellites remain significantly\u0000limited. Motivated by these challenges, we propose a rate-matching framework\u0000for non-orthogonal unicast and multicast (NOUM) transmission. Our approach aims\u0000to minimize the difference between offered rates and traffic demands for both\u0000unicast and multicast messages. By multiplexing unicast and multicast\u0000transmissions over the same radio resource, rate-splitting multiple access\u0000(RSMA) is employed to manage interference between unicast and multicast\u0000streams, as well as inter-user interference under imperfect channel state\u0000information at the LEO satellite. To address the formulated problems\u0000non-smoothness and non-convexity, the common rate is approximated using the\u0000LogSumExp technique. Thereafter, we represent the common rate portion as the\u0000ratio of the approximated function, converting the problem into an\u0000unconstrained form. A generalized power iteration (GPI)-based algorithm, coined\u0000GPI-RS-NOUM, is proposed upon this reformulation. Through comprehensive\u0000numerical analysis across diverse simulation setups, we demonstrate that the\u0000proposed framework outperforms various benchmarks for LEO SATCOM with uneven\u0000traffic demands.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we explore constructions for codes that correct two bursts of deletions, with each burst having length exactly $b$. Previously, the best known construction, derived using the syndrome compression technique, achieved a redundancy of at most $7log n+Oleft(log n/loglog nright)$. In this work, we present new constructions for all $qge 2$ that achieve redundancy at most $7log n+O(loglog n)$ when $b>1$. Additionally, for $b=1$, we provide a new construction of $q$-ary two-deletion correcting codes with redundancy $5log n+O(loglog n)$ for all $q>2$.
{"title":"Codes Correcting Two Bursts of Exactly $b$ Deletions","authors":"Zuo Ye, Wenjun Yu, Ohad Elishco","doi":"arxiv-2408.03113","DOIUrl":"https://doi.org/arxiv-2408.03113","url":null,"abstract":"In this paper, we explore constructions for codes that correct two bursts of\u0000deletions, with each burst having length exactly $b$. Previously, the best\u0000known construction, derived using the syndrome compression technique, achieved\u0000a redundancy of at most $7log n+Oleft(log n/loglog nright)$. In this\u0000work, we present new constructions for all $qge 2$ that achieve redundancy at\u0000most $7log n+O(loglog n)$ when $b>1$. Additionally, for $b=1$, we provide a\u0000new construction of $q$-ary two-deletion correcting codes with redundancy\u0000$5log n+O(loglog n)$ for all $q>2$.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mengyuan Zhao, Maël Le Treust, Tobias J. Oechtering
We study the continuous vector-valued Witsenhausen counterexample through the lens of empirical coordination coding. We characterize the region of achievable pairs of costs in three scenarios: (i) causal encoding and causal decoding, (ii) causal encoding and causal decoding with channel feedback, and (iii) causal encoding and noncausal decoding with channel feedback. In these vector-valued versions of the problem, the optimal coding schemes must rely on a time-sharing strategy, since the region of achievable pairs of costs might not be convex in the scalar version of the problem. We examine the role of the channel feedback when the encoder is causal and the decoder is either causal or non-causal, and we show that feedback improves the performance, only when the decoder is non-causal.
{"title":"Causal Vector-valued Witsenhausen Counterexamples with Feedback","authors":"Mengyuan Zhao, Maël Le Treust, Tobias J. Oechtering","doi":"arxiv-2408.03037","DOIUrl":"https://doi.org/arxiv-2408.03037","url":null,"abstract":"We study the continuous vector-valued Witsenhausen counterexample through the\u0000lens of empirical coordination coding. We characterize the region of achievable\u0000pairs of costs in three scenarios: (i) causal encoding and causal decoding,\u0000(ii) causal encoding and causal decoding with channel feedback, and (iii)\u0000causal encoding and noncausal decoding with channel feedback. In these\u0000vector-valued versions of the problem, the optimal coding schemes must rely on\u0000a time-sharing strategy, since the region of achievable pairs of costs might\u0000not be convex in the scalar version of the problem. We examine the role of the\u0000channel feedback when the encoder is causal and the decoder is either causal or\u0000non-causal, and we show that feedback improves the performance, only when the\u0000decoder is non-causal.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Iván Blanco-Chacón, Alberto F. Boix, Marcus Grefferath, Erik Hieta-aho
The purpose of this paper is to study rank metric codes over Galois rings. In particular, we prove MacWilliams identities for finite chain rings relating the sequences of $q$-binomial moments of a code and its dual and for Gabidulin analogues of free codes over Galois rings we prove the corresponding MacWilliams identity as a functional expression relating the weight enumerator of the dual with the MacWilliams transform of the weight enumerator of the original code plus one extra term attached to the zero divisors.
{"title":"MacWilliams Identities for Rank metric codes over Galois rings","authors":"Iván Blanco-Chacón, Alberto F. Boix, Marcus Grefferath, Erik Hieta-aho","doi":"arxiv-2408.03237","DOIUrl":"https://doi.org/arxiv-2408.03237","url":null,"abstract":"The purpose of this paper is to study rank metric codes over Galois rings. In\u0000particular, we prove MacWilliams identities for finite chain rings relating the\u0000sequences of $q$-binomial moments of a code and its dual and for Gabidulin\u0000analogues of free codes over Galois rings we prove the corresponding\u0000MacWilliams identity as a functional expression relating the weight enumerator\u0000of the dual with the MacWilliams transform of the weight enumerator of the\u0000original code plus one extra term attached to the zero divisors.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The one-day workshop, held prior to the "ZIF Workshop on Information Theory and Related Fields", provided an excellent opportunity for in-depth discussions on several topics within the field of post-Shannon theory. The agenda covered deterministic and randomized identification, focusing on various methods and algorithms for identifying data or signals deterministically and through randomized processes. It explored the theoretical foundations and practical applications of these techniques. The session on resources for increasing identification capacity examined the different resources and strategies that can be utilized to boost the capacity for identifying information. This included discussions on both hardware and software solutions, as well as innovative approaches to resource allocation and optimization. Participants delved into common randomness generation, essential for various cryptographic protocols and communication systems. The session highlighted recent advancements and practical implementations of common randomness in secure communications. The workshop concluded with a detailed look at the development and practical deployment of identification codes. Experts shared insights on code construction techniques, implementation challenges, and real-world applications in various communication systems. We extend our thanks to the esteemed speakers for their valuable contributions: Caspar von Lengerke, Wafa Labidi, Ilya Vorobyev, Johannes Rosenberger, Jonathan Huffmann, and Pau Colomer. Their presentations and insights significantly enriched the workshop. Additionally, we are grateful to all the participants whose active engagement, constructive comments, and stimulating discussions made the event a success. Your involvement was crucial in fostering a collaborative and intellectually vibrant environment.
{"title":"Second 6G life Workshop on Post Shannon Theory","authors":"Yaning Zhao, Christian Deppe","doi":"arxiv-2408.02446","DOIUrl":"https://doi.org/arxiv-2408.02446","url":null,"abstract":"The one-day workshop, held prior to the \"ZIF Workshop on Information Theory\u0000and Related Fields\", provided an excellent opportunity for in-depth discussions\u0000on several topics within the field of post-Shannon theory. The agenda covered\u0000deterministic and randomized identification, focusing on various methods and\u0000algorithms for identifying data or signals deterministically and through\u0000randomized processes. It explored the theoretical foundations and practical\u0000applications of these techniques. The session on resources for increasing\u0000identification capacity examined the different resources and strategies that\u0000can be utilized to boost the capacity for identifying information. This\u0000included discussions on both hardware and software solutions, as well as\u0000innovative approaches to resource allocation and optimization. Participants\u0000delved into common randomness generation, essential for various cryptographic\u0000protocols and communication systems. The session highlighted recent\u0000advancements and practical implementations of common randomness in secure\u0000communications. The workshop concluded with a detailed look at the development\u0000and practical deployment of identification codes. Experts shared insights on\u0000code construction techniques, implementation challenges, and real-world\u0000applications in various communication systems. We extend our thanks to the\u0000esteemed speakers for their valuable contributions: Caspar von Lengerke, Wafa\u0000Labidi, Ilya Vorobyev, Johannes Rosenberger, Jonathan Huffmann, and Pau\u0000Colomer. Their presentations and insights significantly enriched the workshop.\u0000Additionally, we are grateful to all the participants whose active engagement,\u0000constructive comments, and stimulating discussions made the event a success.\u0000Your involvement was crucial in fostering a collaborative and intellectually\u0000vibrant environment.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the conditional independence (CI) implication problem with bounded cardinalities, which asks whether a given CI implication holds for all discrete random variables with given cardinalities, is co-NEXPTIME-hard. The problem remains co-NEXPTIME-hard if all variables are binary. The reduction goes from a variant of the tiling problem and is based on a prior construction used by Cheuk Ting Li to show the undecidability of a related problem where the cardinality of some variables remains unbounded. The CI implication problem with bounded cardinalities is known to be in EXPSPACE, as its negation can be stated as an existential first-order logic formula over the reals of size exponential with regard to the size of the input.
我们证明，有界万有引力的条件独立性（CI）蕴涵问题（该问题询问给定的 CI 蕴涵对于给定万有引力的所有离散随机变量是否成立）是共 NEXPTIME 难问题。如果所有变量都是二进制变量，这个问题仍然是共 NEXPTIME-hard。这一还原源于平铺问题的一个变体，并基于李卓廷用来证明某些变量的卡方根性仍然无界的相关问题的不可判定性的先验构造。众所周知，有界心数的 CI 含义问题在 EXPSPACE 中，因为它的否定可以最好地表示为一个存在的一阶逻辑公式，这个公式在有界心数的实数上，其大小与输入的大小成指数关系。
{"title":"On the Complexity of the Conditional Independence Implication Problem With Bounded Cardinalities","authors":"Michał Makowski","doi":"arxiv-2408.02550","DOIUrl":"https://doi.org/arxiv-2408.02550","url":null,"abstract":"We show that the conditional independence (CI) implication problem with\u0000bounded cardinalities, which asks whether a given CI implication holds for all\u0000discrete random variables with given cardinalities, is co-NEXPTIME-hard. The\u0000problem remains co-NEXPTIME-hard if all variables are binary. The reduction\u0000goes from a variant of the tiling problem and is based on a prior construction\u0000used by Cheuk Ting Li to show the undecidability of a related problem where the\u0000cardinality of some variables remains unbounded. The CI implication problem\u0000with bounded cardinalities is known to be in EXPSPACE, as its negation can be\u0000stated as an existential first-order logic formula over the reals of size\u0000exponential with regard to the size of the input.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sudhir R. Ghorpade, Trygve Johnsen, Rati Ludhani, Rakhi Pratihar
We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$ of finite degree. To obtain our results we will utilize well-known connections between these weights and properties of the Stanley-Reisner rings of a series of matroids associated to each code $C_q$. In the process, we are able to explicitly determine all the graded Betti numbers of matroids associated to $C_q$ and its elongations.
{"title":"Higher weight spectra and Betti numbers of Reed-Muller codes $RM_2(2,2)$","authors":"Sudhir R. Ghorpade, Trygve Johnsen, Rati Ludhani, Rakhi Pratihar","doi":"arxiv-2408.02548","DOIUrl":"https://doi.org/arxiv-2408.02548","url":null,"abstract":"We determine the higher weight spectra of $q$-ary Reed-Muller codes\u0000$C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the\u0000usual weight distributions of all extension codes of $C_q$ over every field\u0000extension of $F_q$ of finite degree. To obtain our results we will utilize\u0000well-known connections between these weights and properties of the\u0000Stanley-Reisner rings of a series of matroids associated to each code $C_q$. In\u0000the process, we are able to explicitly determine all the graded Betti numbers\u0000of matroids associated to $C_q$ and its elongations.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In recent literature, matrix product (MP) codes and their duals have gained significant attention due to their application in the construction of quantum stabilizer codes. In this paper, we begin with providing a formula that characterizes the Galois dual of MP codes. Using this formula, we establish the conditions under which MP codes are self-orthogonal and dual-containing. Although similar results may exist in the literature, the novelty and superiority of our results can be identified in the following points. Previous results that characterize the duals of MP codes only apply to MP codes with an invertible square defining matrix $mathcal{A}$. However, our characterization applies to MP code with any defining matrix, whether $mathcal{A}$ is not square or not of full row rank. Previous studies on the conditions for self-orthogonality or dual-containment of MP codes have assumed certain structures for the product $mathcal{A}mathcal{A}^T$ or $mathcal{A}mathcal{A}^{dagger}$, such as being diagonal, anti-diagonal, monomial, or partitioned Hermitian orthogonal. However, our conditions do not necessitate such specific structures. Previous studies investigated MP code duality in the context of Euclidean and Hermitian duals; however, we investigate MP code duality in the broader context of Galois dual, with Euclidean and Hermitian duals acting as special cases. Finally, it is worth noting that the proposed conditions for Galois self-orthogonality or dual-containment are both necessary and sufficient. To demonstrate the theoretical results, several numerical examples with best-known parameters MP codes are provided.
{"title":"On Galois duality, self-orthogonality, and dual-containment of matrix product codes","authors":"Ramy Farouk Taki Eldin","doi":"arxiv-2408.02799","DOIUrl":"https://doi.org/arxiv-2408.02799","url":null,"abstract":"In recent literature, matrix product (MP) codes and their duals have gained\u0000significant attention due to their application in the construction of quantum\u0000stabilizer codes. In this paper, we begin with providing a formula that\u0000characterizes the Galois dual of MP codes. Using this formula, we establish the\u0000conditions under which MP codes are self-orthogonal and dual-containing.\u0000Although similar results may exist in the literature, the novelty and\u0000superiority of our results can be identified in the following points. Previous\u0000results that characterize the duals of MP codes only apply to MP codes with an\u0000invertible square defining matrix $mathcal{A}$. However, our characterization\u0000applies to MP code with any defining matrix, whether $mathcal{A}$ is not\u0000square or not of full row rank. Previous studies on the conditions for\u0000self-orthogonality or dual-containment of MP codes have assumed certain\u0000structures for the product $mathcal{A}mathcal{A}^T$ or\u0000$mathcal{A}mathcal{A}^{dagger}$, such as being diagonal, anti-diagonal,\u0000monomial, or partitioned Hermitian orthogonal. However, our conditions do not\u0000necessitate such specific structures. Previous studies investigated MP code\u0000duality in the context of Euclidean and Hermitian duals; however, we\u0000investigate MP code duality in the broader context of Galois dual, with\u0000Euclidean and Hermitian duals acting as special cases. Finally, it is worth\u0000noting that the proposed conditions for Galois self-orthogonality or\u0000dual-containment are both necessary and sufficient. To demonstrate the\u0000theoretical results, several numerical examples with best-known parameters MP\u0000codes are provided.","PeriodicalId":501082,"journal":{"name":"arXiv - MATH - Information Theory","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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