Pub Date : 2024-11-20DOI: 10.1109/TIT.2024.3503500
Adam Wills;Ting-Chun Lin;Min-Hsiu Hsieh
In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings’ weight reduction construction for qLDPC codes to give a weight reduction procedure for qLTCs. Secondly, we describe a novel ‘soundness amplification’ procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code and the hemicubic code yield many previously unknown parameters. In addition, applications of all three results are described to an upcoming work.
{"title":"Tradeoff Constructions for Quantum Locally Testable Codes","authors":"Adam Wills;Ting-Chun Lin;Min-Hsiu Hsieh","doi":"10.1109/TIT.2024.3503500","DOIUrl":"https://doi.org/10.1109/TIT.2024.3503500","url":null,"abstract":"In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings’ weight reduction construction for qLDPC codes to give a weight reduction procedure for qLTCs. Secondly, we describe a novel ‘soundness amplification’ procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code and the hemicubic code yield many previously unknown parameters. In addition, applications of all three results are described to an upcoming work.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"426-458"},"PeriodicalIF":2.2,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10759074","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1109/TIT.2024.3500578
Joseph M. Renes
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constant $E(W,R)$ which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.
{"title":"Tight Lower Bound on the Error Exponent of Classical-Quantum Channels","authors":"Joseph M. Renes","doi":"10.1109/TIT.2024.3500578","DOIUrl":"https://doi.org/10.1109/TIT.2024.3500578","url":null,"abstract":"A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constant \u0000<inline-formula> <tex-math>$E(W,R)$ </tex-math></inline-formula>\u0000 which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"530-538"},"PeriodicalIF":2.2,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1109/TIT.2024.3502295
Nicholas R. Olson;Jeffrey G. Andrews
We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this as the matrix Laplace transform and establish that it is in general a matrix function extension of the scalar Laplace transform. We show that the matrix Laplace transform of Poisson shot noise admits an expression analogous to that implied by Campbell’s theorem. We demonstrate the utility of this generalization of Campbell’s theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary CDF (CCDF) and meta-distribution of signal-to-interference-and-noise (SINR) models in Poisson networks. In the former application, we demonstrate how the higher order moments of Poisson shot noise may be obtained directly from the elements of its matrix Laplace transform. We further show how the CCDF of this object may be bounded using a summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed desired signal power may be obtained via an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading. Additionally, when the power of the desired signal is exponentially distributed, we establish that the meta-distribution may be obtained in terms of the limit of a sequence expressed in terms of the matrix Laplace transform of a related Poisson shot noise process.
{"title":"A Matrix Exponential Generalization of the Laplace Transform of Poisson Shot Noise","authors":"Nicholas R. Olson;Jeffrey G. Andrews","doi":"10.1109/TIT.2024.3502295","DOIUrl":"https://doi.org/10.1109/TIT.2024.3502295","url":null,"abstract":"We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this as the matrix Laplace transform and establish that it is in general a matrix function extension of the scalar Laplace transform. We show that the matrix Laplace transform of Poisson shot noise admits an expression analogous to that implied by Campbell’s theorem. We demonstrate the utility of this generalization of Campbell’s theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary CDF (CCDF) and meta-distribution of signal-to-interference-and-noise (SINR) models in Poisson networks. In the former application, we demonstrate how the higher order moments of Poisson shot noise may be obtained directly from the elements of its matrix Laplace transform. We further show how the CCDF of this object may be bounded using a summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed desired signal power may be obtained via an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading. Additionally, when the power of the desired signal is exponentially distributed, we establish that the meta-distribution may be obtained in terms of the limit of a sequence expressed in terms of the matrix Laplace transform of a related Poisson shot noise process.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"396-412"},"PeriodicalIF":2.2,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1109/TIT.2024.3493762
Utkarsh Gupta;Hessam Mahdavifar
In this paper, we propose a new algorithm for the problem of string reconstruction from its substring composition multiset. Motivated by applications in polymer-based data storage for recovering strings from tandem mass-spectrometry sequencing, the proposed algorithm leverages the equivalent polynomial formulation of the problem which facilitates efficient parallel implementation. The computational complexity of the proposed reconstruction algorithm is upper bounded by $6.5n^{2}$ finite field operations, where the field size is upper bounded by $10n$ , implying that the computational complexity is upper bounded by $6.5n^{2}(3.22+log {n})$ binary operations. Furthermore, it allows parallelization leading to $O(n log n)$ reconstruction latency. We characterize sufficient conditions for a length n binary string that guarantee the string’s reconstruction time complexity to be bounded polynomially. Moreover, the sufficient conditions on binary strings that guarantee reconstruction in polynomial time are more general than the conditions for the algorithm by Acharya et al. This is used to construct new codebooks of reconstruction codes that have efficient encoding procedures, and are larger, by at least a linear factor in size, compared to the previously best known construction by Pattabiraman et al., (2023).
{"title":"A New Algebraic Approach for String Reconstruction From Substring Compositions","authors":"Utkarsh Gupta;Hessam Mahdavifar","doi":"10.1109/TIT.2024.3493762","DOIUrl":"https://doi.org/10.1109/TIT.2024.3493762","url":null,"abstract":"In this paper, we propose a new algorithm for the problem of string reconstruction from its substring composition multiset. Motivated by applications in polymer-based data storage for recovering strings from tandem mass-spectrometry sequencing, the proposed algorithm leverages the equivalent polynomial formulation of the problem which facilitates efficient parallel implementation. The computational complexity of the proposed reconstruction algorithm is upper bounded by \u0000<inline-formula> <tex-math>$6.5n^{2}$ </tex-math></inline-formula>\u0000 finite field operations, where the field size is upper bounded by \u0000<inline-formula> <tex-math>$10n$ </tex-math></inline-formula>\u0000, implying that the computational complexity is upper bounded by \u0000<inline-formula> <tex-math>$6.5n^{2}(3.22+log {n})$ </tex-math></inline-formula>\u0000 binary operations. Furthermore, it allows parallelization leading to \u0000<inline-formula> <tex-math>$O(n log n)$ </tex-math></inline-formula>\u0000 reconstruction latency. We characterize sufficient conditions for a length n binary string that guarantee the string’s reconstruction time complexity to be bounded polynomially. Moreover, the sufficient conditions on binary strings that guarantee reconstruction in polynomial time are more general than the conditions for the algorithm by Acharya et al. This is used to construct new codebooks of reconstruction codes that have efficient encoding procedures, and are larger, by at least a linear factor in size, compared to the previously best known construction by Pattabiraman et al., (2023).","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"125-137"},"PeriodicalIF":2.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1109/TIT.2024.3498033
Han Cai;Ying Miao;Moshe Schwartz;Xiaohu Tang
In this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.n this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.
{"title":"Repairing Schemes for Tamo-Barg Codes","authors":"Han Cai;Ying Miao;Moshe Schwartz;Xiaohu Tang","doi":"10.1109/TIT.2024.3498033","DOIUrl":"https://doi.org/10.1109/TIT.2024.3498033","url":null,"abstract":"In this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.n this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"227-243"},"PeriodicalIF":2.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1109/TIT.2024.3497920
Hua Sun
In the secure groupcast problem, a transmitter wants to securely groupcast a message with the maximum rate to the first N of K receivers by broadcasting with the minimum bandwidth, where the K receivers are each equipped with a key variable from a known joint distribution. Examples are provided to prove that different instances of secure groupcast that have the same entropic structure, i.e., the same entropy for all subsets of the key variables, can have different maximum groupcast rates and different minimum broadcast bandwidth. Thus, extra-entropic structure matters for secure groupcast. Next, the maximum groupcast rate is explored when the key variables are generic linear combinations of a basis set of independent key symbols, i.e., the keys lie in generic subspaces. The maximum groupcast rate is characterized when the dimension of each key subspace is either small or large, i.e., the extreme regimes. For the intermediate regime, various interference alignment schemes originated from wireless interference networks, such as eigenvector based and asymptotic schemes, are shown to be useful.
{"title":"Secure Groupcast: Extra-Entropic Structure and Linear Feasibility","authors":"Hua Sun","doi":"10.1109/TIT.2024.3497920","DOIUrl":"https://doi.org/10.1109/TIT.2024.3497920","url":null,"abstract":"In the secure groupcast problem, a transmitter wants to securely groupcast a message with the maximum rate to the first N of K receivers by broadcasting with the minimum bandwidth, where the K receivers are each equipped with a key variable from a known joint distribution. Examples are provided to prove that different instances of secure groupcast that have the same entropic structure, i.e., the same entropy for all subsets of the key variables, can have different maximum groupcast rates and different minimum broadcast bandwidth. Thus, extra-entropic structure matters for secure groupcast. Next, the maximum groupcast rate is explored when the key variables are generic linear combinations of a basis set of independent key symbols, i.e., the keys lie in generic subspaces. The maximum groupcast rate is characterized when the dimension of each key subspace is either small or large, i.e., the extreme regimes. For the intermediate regime, various interference alignment schemes originated from wireless interference networks, such as eigenvector based and asymptotic schemes, are shown to be useful.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"683-697"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of designing explicit bent and plateaued functions has been researched for several decades. However, finding new bent functions outside the well-known completed Maiorana-McFarland class $mathcal {M}^{#}$ is still a challenge. Plateaued functions have been characterized in many different ways, but there is no general and rigorous mathematical method to generate them directly, except for the ones in the spirit of the well-known Maiorana-McFarland constructions or those obtained through adaptations of the secondary constructions of bent functions. Jeong and Lee recently made significant advances regarding algorithms for constructing balanced plateaued functions with maximal algebraic degrees in [IEEE Trans. Inf. Theory, 70(2), 1408-1421, 2024]. Due to the gap between our significant interest in the notion of plateaued functions and the knowledge we have on it, our motivation is to bring further results on the constructions of plateaued functions that allow us to understand their structure better. This article creates a framework of new generic constructions of bent and plateaued functions by studying Boolean functions of the form $h(x)=f(x)+F(f_{1}(x),ldots, f_{r}(x))$ , where $f_{i}(x)=f(x)+f(x+mu _{i})$ for each $1leq ileq r$ . We firstly prove that h and f have the same extended Walsh-Hadamard spectrum if $D_{mu _{i}}D_{mu _{j}}f=0$ for any $1leq ilt jleq r$ . This result extends a previous construction of bent functions to any Boolean functions. The strength of such a result is that it allows us to obtain several plateaued functions of high algebraic degrees from known ones with low algebraic degrees, which was a significant and challenging problem raised in the literature. Such a result is a real challenge and breaks a deadlock since no mathematical method allows the general constructions of plateaued functions. We next give an extended affine equivalent form of the function h, which provides us with another compelling perspective to design new bent functions (including those which are outside $mathcal {M}^{#}$ from certain known ones inside $mathcal {M}^{#}$ ) and plateaued functions. Finally, we present four generic constructions of bent functions outside $mathcal {M}^{#}$ from generalized Maiorana-McFarland functions.
{"title":"Direct Approaches for Generic Constructions of Plateaued Functions and Bent Functions Outside M#","authors":"Yanjun Li;Haibin Kan;Sihem Mesnager;Jie Peng;Lijing Zheng","doi":"10.1109/TIT.2024.3497804","DOIUrl":"https://doi.org/10.1109/TIT.2024.3497804","url":null,"abstract":"The problem of designing explicit bent and plateaued functions has been researched for several decades. However, finding new bent functions outside the well-known completed Maiorana-McFarland class <inline-formula> <tex-math>$mathcal {M}^{#}$ </tex-math></inline-formula> is still a challenge. Plateaued functions have been characterized in many different ways, but there is no general and rigorous mathematical method to generate them directly, except for the ones in the spirit of the well-known Maiorana-McFarland constructions or those obtained through adaptations of the secondary constructions of bent functions. Jeong and Lee recently made significant advances regarding algorithms for constructing balanced plateaued functions with maximal algebraic degrees in [IEEE Trans. Inf. Theory, 70(2), 1408-1421, 2024]. Due to the gap between our significant interest in the notion of plateaued functions and the knowledge we have on it, our motivation is to bring further results on the constructions of plateaued functions that allow us to understand their structure better. This article creates a framework of new generic constructions of bent and plateaued functions by studying Boolean functions of the form <inline-formula> <tex-math>$h(x)=f(x)+F(f_{1}(x),ldots, f_{r}(x))$ </tex-math></inline-formula>, where <inline-formula> <tex-math>$f_{i}(x)=f(x)+f(x+mu _{i})$ </tex-math></inline-formula> for each <inline-formula> <tex-math>$1leq ileq r$ </tex-math></inline-formula>. We firstly prove that h and f have the same extended Walsh-Hadamard spectrum if <inline-formula> <tex-math>$D_{mu _{i}}D_{mu _{j}}f=0$ </tex-math></inline-formula> for any <inline-formula> <tex-math>$1leq ilt jleq r$ </tex-math></inline-formula>. This result extends a previous construction of bent functions to any Boolean functions. The strength of such a result is that it allows us to obtain several plateaued functions of high algebraic degrees from known ones with low algebraic degrees, which was a significant and challenging problem raised in the literature. Such a result is a real challenge and breaks a deadlock since no mathematical method allows the general constructions of plateaued functions. We next give an extended affine equivalent form of the function h, which provides us with another compelling perspective to design new bent functions (including those which are outside <inline-formula> <tex-math>$mathcal {M}^{#}$ </tex-math></inline-formula> from certain known ones inside <inline-formula> <tex-math>$mathcal {M}^{#}$ </tex-math></inline-formula>) and plateaued functions. Finally, we present four generic constructions of bent functions outside <inline-formula> <tex-math>$mathcal {M}^{#}$ </tex-math></inline-formula> from generalized Maiorana-McFarland functions.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 2","pages":"1400-1418"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143361258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1109/TIT.2024.3497008
Chaofeng Guan;Ruihu Li;Jingjie Lv;Zhi Ma
In this paper, we establish the necessary and sufficient conditions for quasi-cyclic (QC) codes with index even to be symplectic self-orthogonal. Subsequently, we present the lower and upper bounds on the minimum symplectic distances of a class of 1-generator QC codes and their symplectic dual codes by decomposing code spaces. As an application, we construct many new binary symplectic self-orthogonal QC codes with excellent parameters, leading to 117 record-breaking quantum error-correction codes.
{"title":"Symplectic Self-Orthogonal Quasi-Cyclic Codes","authors":"Chaofeng Guan;Ruihu Li;Jingjie Lv;Zhi Ma","doi":"10.1109/TIT.2024.3497008","DOIUrl":"https://doi.org/10.1109/TIT.2024.3497008","url":null,"abstract":"In this paper, we establish the necessary and sufficient conditions for quasi-cyclic (QC) codes with index even to be symplectic self-orthogonal. Subsequently, we present the lower and upper bounds on the minimum symplectic distances of a class of 1-generator QC codes and their symplectic dual codes by decomposing code spaces. As an application, we construct many new binary symplectic self-orthogonal QC codes with excellent parameters, leading to 117 record-breaking quantum error-correction codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"114-124"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1109/TIT.2024.3496486
Jamison R. Ebert;Jean-Francois Chamberland;Krishna R. Narayanan
This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC), whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this BP denoiser falls within the framework of non-separable denoising functions and subsequently, that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means for obtaining shaping gains over the AWGN channel.
{"title":"Sparse Regression LDPC Codes","authors":"Jamison R. Ebert;Jean-Francois Chamberland;Krishna R. Narayanan","doi":"10.1109/TIT.2024.3496486","DOIUrl":"https://doi.org/10.1109/TIT.2024.3496486","url":null,"abstract":"This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC), whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this BP denoiser falls within the framework of non-separable denoising functions and subsequently, that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means for obtaining shaping gains over the AWGN channel.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"167-191"},"PeriodicalIF":2.2,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although the expenses associated with DNA sequencing have been rapidly decreasing, the current cost of sequencing information stands at roughly ${$}120$ /GB, which is dramatically more expensive than reading from existing archival storage solutions today. In this work, we aim to reduce not only the cost but also the latency of DNA storage by initiating the study of the DNA coverage depth problem, which aims to reduce the required number of reads to retrieve information from the storage system. Under this framework, our main goal is to understand the effect of error-correcting codes and retrieval algorithms on the required sequencing coverage depth. We establish that the expected number of reads that are required for information retrieval is minimized when the channel follows a uniform distribution. We also derive upper and lower bounds on the probability distribution of this number of required reads and provide a comprehensive upper and lower bound on its expected value. We further prove that for a noiseless channel and uniform distribution, MDS codes are optimal in terms of minimizing the expected number of reads. Additionally, we study the DNA coverage depth problem under the random-access setup, in which the user aims to retrieve just a specific information unit from the entire DNA storage system. We prove that the expected retrieval time is at least k for $[n,k]$ MDS codes as well as for other families of codes. Furthermore, we present explicit code constructions that achieve expected retrieval times below k and evaluate their performance through analytical methods and simulations. Lastly, we provide lower bounds on the maximum expected retrieval time. Our findings offer valuable insights for reducing the cost and latency of DNA storage.
{"title":"Cover Your Bases: How to Minimize the Sequencing Coverage in DNA Storage Systems","authors":"Daniella Bar-Lev;Omer Sabary;Ryan Gabrys;Eitan Yaakobi","doi":"10.1109/TIT.2024.3496587","DOIUrl":"https://doi.org/10.1109/TIT.2024.3496587","url":null,"abstract":"Although the expenses associated with DNA sequencing have been rapidly decreasing, the current cost of sequencing information stands at roughly \u0000<inline-formula> <tex-math>${$}120$ </tex-math></inline-formula>\u0000/GB, which is dramatically more expensive than reading from existing archival storage solutions today. In this work, we aim to reduce not only the cost but also the latency of DNA storage by initiating the study of the DNA coverage depth problem, which aims to reduce the required number of reads to retrieve information from the storage system. Under this framework, our main goal is to understand the effect of error-correcting codes and retrieval algorithms on the required sequencing coverage depth. We establish that the expected number of reads that are required for information retrieval is minimized when the channel follows a uniform distribution. We also derive upper and lower bounds on the probability distribution of this number of required reads and provide a comprehensive upper and lower bound on its expected value. We further prove that for a noiseless channel and uniform distribution, MDS codes are optimal in terms of minimizing the expected number of reads. Additionally, we study the DNA coverage depth problem under the random-access setup, in which the user aims to retrieve just a specific information unit from the entire DNA storage system. We prove that the expected retrieval time is at least k for \u0000<inline-formula> <tex-math>$[n,k]$ </tex-math></inline-formula>\u0000 MDS codes as well as for other families of codes. Furthermore, we present explicit code constructions that achieve expected retrieval times below k and evaluate their performance through analytical methods and simulations. Lastly, we provide lower bounds on the maximum expected retrieval time. Our findings offer valuable insights for reducing the cost and latency of DNA storage.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"192-218"},"PeriodicalIF":2.2,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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