IEEE Transactions on Information Theory最新文献

{"title":"Channel Coding for Gaussian Channels With Multifaceted Power Constraints","authors":"Adeel Mahmood;Aaron B. Wagner","doi":"10.1109/TIT.2026.3690380","DOIUrl":"https://doi.org/10.1109/TIT.2026.3690380","url":null,"abstract":"Through refined asymptotic analysis based on the normal approximation, we study how higher-order coding performance depends on the mean power as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary (but finite) number of arbitrary functions of the normalized average power is constrained. The framework generalizes existing models, recovering the standard maximal and expected power constraints and the recent mean and variance constraint as special cases. Under certain growth and continuity assumptions on the functions, our main theorem gives an exact characterization of the minimum average error probability for Gaussian channels as a function of the first- and second-order coding rates. The converse proof reduces the code design problem to minimization over a compact (under the Prokhorov metric) set of probability distributions, characterizes the extreme points of this set and invokes the Bauer’s maximization principle. Our results for the multifaceted power model serve as more precise benchmarks for practical modulation schemes with multiple amplitude levels, probabilistic shaping and nonuniform constellation geometries.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 7","pages":"4602-4617"},"PeriodicalIF":2.6,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148504452","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}

{"title":"Improved Constructions of Linear Codes for Insertions and Deletions","authors":"Roee Gross;Roni Con;Eitan Yaakobi","doi":"10.1109/TIT.2026.3678200","DOIUrl":"https://doi.org/10.1109/TIT.2026.3678200","url":null,"abstract":"In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and decoding algorithms, making them highly desirable. A key challenge in this area is achieving rates close to the half-Singleton bound for efficient linear codes over finite fields. We improve upon previous results by constructing explicit codes over <inline-formula> <tex-math>$mathbb {F}_{q^{2}}$ </tex-math></inline-formula>, linear over <inline-formula> <tex-math>$mathbb {F}_{q}$ </tex-math></inline-formula>, with rate <inline-formula> <tex-math>$1/2 - delta - varepsilon $ </tex-math></inline-formula> that can efficiently correct a <inline-formula> <tex-math>$delta $ </tex-math></inline-formula>-fraction of indel errors, where <inline-formula> <tex-math>$q = O(varepsilon ^{-4})$ </tex-math></inline-formula>. Additionally, we construct fully linear codes over <inline-formula> <tex-math>$mathbb {F}_{q}$ </tex-math></inline-formula> with rate <inline-formula> <tex-math>$1/2 - 2sqrt {delta } - varepsilon $ </tex-math></inline-formula> that can also efficiently correct <inline-formula> <tex-math>$delta $ </tex-math></inline-formula>-fraction of indels. These results significantly advance the study of linear codes for the indel model, bringing them closer to the theoretical half-Singleton bound. We also generalize the half-Singleton bound, for every code <inline-formula> <tex-math>$C subseteq mathbb {F}^{n}$ </tex-math></inline-formula> linear over <inline-formula> <tex-math>$mathbb {E} subset mathbb {F}$ </tex-math></inline-formula> a subfield of <inline-formula> <tex-math>$mathbb {F}$ </tex-math></inline-formula>, such that <inline-formula> <tex-math>$C$ </tex-math></inline-formula> has the ability to correct <inline-formula> <tex-math>$delta $ </tex-math></inline-formula>-fraction of indels, the rate is bounded by <inline-formula> <tex-math>$(1-delta)/2$ </tex-math></inline-formula>.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3830-3839"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509731","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}

{"title":"Cyclic Scheduler Design for Minimizing Age of Information in Massive Scale Networks Susceptible to Packet Errors","authors":"Sahan Liyanaarachchi;Sennur Ulukus;Nail Akar","doi":"10.1109/TIT.2025.3548017","DOIUrl":"https://doi.org/10.1109/TIT.2025.3548017","url":null,"abstract":"In multi-source status update systems, sources need to be scheduled appropriately to maintain timely communication between each of the sources and the monitor. A <italic>cyclic schedule</i> is an age-agnostic schedule in which the sources are served according to a fixed finite transmission pattern, which upon completion, repeats itself. Such a scheme has a low <inline-formula> <tex-math>$O(1)$ </tex-math></inline-formula> runtime complexity, which is desirable in large networks. This paper’s focus is on designing transmission patterns so as to be used in massive scale networking scenarios involving a very large number of sources, e.g., up to thousands of IoT sources, with service time requirements and weights being heterogeneous in nature. The goal is to minimize the weighted sum age of information (AoI), called weighted AoI, when transmitting users’ packets over a channel susceptible to heterogeneous packet errors. The main tool we use is a stochastic modeling framework using either Markov chains (MC) or moment generating functions (MGF), by which we obtain the weighted AoI for a given transmission pattern, which is not straightforward in the presence of packet drops. Using this framework, we provide a lower bound on the weighted AoI for the particular case of two sources, and also an algorithm to attain this lower bound. Then, by using the same framework, we design a cyclic scheduler for general number of sources with reasonable complexity using convex optimization and well-established packet spreading algorithms, and comparatively evaluate the proposed algorithm and existing age-agnostic scheduling schemes for general number of sources (resp. two sources) when the lower bound is not available (resp. when it is available). We present extensive numerical results also involving comparisons with age-aware scheduling to validate the effectiveness of the proposed approach.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3927-3949"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509777","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}

{"title":"An Open Problem on NSC Quasi-Unitary Matrices and Its Application to Quantum Codes","authors":"Meng Cao;Kun Zhou","doi":"10.1109/TIT.2026.3681560","DOIUrl":"https://doi.org/10.1109/TIT.2026.3681560","url":null,"abstract":"Matrix-product (MP) codes provide an efficient approach for constructing long classical codes by combining a defining matrix with several shorter constituent codes. Employing non-singular by columns (NSC) quasi-unitary matrices as the defining matrices of MP codes enables the construction of various types of quantum codes. The existence of infinite families of <inline-formula> <tex-math>$Ntimes N$ </tex-math></inline-formula> NSC quasi-unitary matrices over <inline-formula> <tex-math>$mathbb {F}_{q^{2}}$ </tex-math></inline-formula> for <inline-formula> <tex-math>$N=q$ </tex-math></inline-formula> or <inline-formula> <tex-math>$q+2 leq N leq q^{2}$ </tex-math></inline-formula>, where <inline-formula> <tex-math>$q neq 2$ </tex-math></inline-formula> is a prime power, was posed as an open problem in the literature and remains largely unresolved. In this paper, we introduce a novel method for constructing NSC quasi-unitary matrices based on a generalized orthogonality relation for cyclic groups. We prove that <inline-formula> <tex-math>$(2m+1)times (2m+1)$ </tex-math></inline-formula> NSC quasi-unitary matrices always exist over <inline-formula> <tex-math>$mathbb {F}_{q^{2}}$ </tex-math></inline-formula> whenever <inline-formula> <tex-math>$2mmid (q^{2}-1)$ </tex-math></inline-formula>. This result establishes, for the first time, the existence of six new infinite families of NSC quasi-unitary matrices, providing solutions to the aforementioned open problem. The number of such matrices within our framework is also determined. As a key application, we leverage NSC quasi-unitary matrices to construct optimal pure quantum <inline-formula> <tex-math>$(r,delta)$ </tex-math></inline-formula>-locally repairable codes (LRCs). We propose an effective method for constructing optimal pure quantum <inline-formula> <tex-math>$(r,delta)$ </tex-math></inline-formula>-LRCs from MP codes whose defining matrices are NSC quasi-unitary matrices. Based on this method, we explicitly construct two new infinite families of optimal pure quantum <inline-formula> <tex-math>$(r,delta)$ </tex-math></inline-formula>-LRCs with flexible parameters.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3885-3898"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148510077","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}

{"title":"Generalization Bounds for Quantum Learning via Rényi Divergences","authors":"Naqueeb Ahmad Warsi;Ayanava Dasgupta;Masahito Hayashi","doi":"10.1109/TIT.2026.3683678","DOIUrl":"https://doi.org/10.1109/TIT.2026.3683678","url":null,"abstract":"This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical Rényi divergences, utilizing a variational approach for evaluating quantum Rényi divergences, specifically the Petz and a newly introduced modified sandwich quantum Rényi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum Rényi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum Rényi divergence and classical Rényi divergence, and another employing smooth max Rényi divergence.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"4012-4040"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511124","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}

{"title":"Byzantine-Eavesdropper Alliance: How to Achieve Symmetric Privacy in Quantum X-Secure B-Byzantine E-Eavesdropped U-Unresponsive T-Colluding PIR?","authors":"Mohamed Nomeir;Alptug Aytekin;Sennur Ulukus","doi":"10.1109/TIT.2025.3623081","DOIUrl":"https://doi.org/10.1109/TIT.2025.3623081","url":null,"abstract":"We consider the quantum <italic>symmetric</i> private information retrieval (QSPIR) problem in a system with <italic>N</i> databases and <italic>K</i> messages, with <italic>U</i> unresponsive servers, <italic>T</i>-colluding servers, and <italic>X</i>-security parameter, under several fundamental threat models. In the first model, there are <inline-formula> <tex-math>$mathcal {E}_{1}$ </tex-math></inline-formula> eavesdropped links in the uplink direction (the direction from the user to the <italic>N</i> servers), <inline-formula> <tex-math>$mathcal {E}_{2}$ </tex-math></inline-formula> eavesdropped links in the downlink direction (the direction from the servers to the user), where <inline-formula> <tex-math>$|mathcal {E}_{1}|, |mathcal {E}_{2}| leq E$ </tex-math></inline-formula>; we coin this eavesdropper setting as <italic>dynamic</i> eavesdroppers. We show that super-dense coding gain can be achieved for some regimes. In the second model, we consider the case with Byzantine servers, i.e., servers that can coordinate to devise a plan to harm the privacy and security of the system together with static eavesdroppers, by listening to the same links in both uplink and downlink directions. It is important to note the considerable difference between the two threat models, since the eavesdroppers can take huge advantage of the presence of the Byzantine servers. Unlike the previous works in SPIR with Byzantine servers, that assume that the Byzantine servers can send only random symbols independent of the stored messages, we follow the original definition of Byzantine servers inPIR proposed by Banawan-Ulukus, where the Byzantine servers can send symbols that can be functions of the storage, queries, as well as the random symbols in a way that can produce worse harm to the system. In the third and the most novel threat model, we consider the presence of Byzantine servers and dynamic eavesdroppers together. We show that having dynamic eavesdroppers along with Byzantine servers in the same system model creates more threats to the system than having static eavesdroppers with Byzantine servers. This is the first work that considers the quantum version of unresponsive and eavesdropped threat model. In addition, this is the first work, classical or quantum, that considers the presence of Byzantine servers together with eavesdroppers in the same system model (static or dynamic eavesdroppers). Another layer of difficulty that we handle in this work stems from the <italic>symmetric</i> privacy requirement in the presence of Byzantine servers, which by itself has never been studied before in classical or quantum variations; here the Byzantine servers may attempt to leak information about undesired messages to the user, which is not allowed in SPIR.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"4312-4328"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511790","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}

{"title":"A New Construction Structure on Coded Caching With Linear Subpacketization: Non-Half-Sum Disjoint Packing","authors":"Minquan Cheng;Huimei Wei;Kai Wan;Giuseppe Caire","doi":"10.1109/TIT.2026.3685007","DOIUrl":"https://doi.org/10.1109/TIT.2026.3685007","url":null,"abstract":"Coded caching is a promising technique for effectively reducing peak traffic by using local caches and the multicast gains generated by these local caches. Coded caching schemes have been widely investigated, following the seminal work of Maddah-Ali and Niesen. Explicit coding constructions have been proposed for a variety of network topologies with information-theoretically optimal or near-optimal transmission load <inline-formula> <tex-math>$R$ </tex-math></inline-formula>. An important parameter in these constructions is the subpacketization <inline-formula> <tex-math>$F$ </tex-math></inline-formula>, i.e., the number of subpackets that each content file needs to be divided into. In particular, the original scheme of Maddah-Ali and Niesen as well as several other variants require <inline-formula> <tex-math>$F$ </tex-math></inline-formula> to grow exponentially with the number of users <inline-formula> <tex-math>$K$ </tex-math></inline-formula>. In practice, files have finite size and too large <inline-formula> <tex-math>$F$ </tex-math></inline-formula> yields impractically small subpackets. Therefore, it is important to design coded caching schemes with <inline-formula> <tex-math>$F$ </tex-math></inline-formula> and <inline-formula> <tex-math>$R$ </tex-math></inline-formula> as small as possible. At present, the few known schemes with subpacketization linear in <inline-formula> <tex-math>$K$ </tex-math></inline-formula> achieve large load. In this paper, we consider the linear scaling regime <inline-formula> <tex-math>$F = O(K)$ </tex-math></inline-formula> and design schemes with a lower transmission load <inline-formula> <tex-math>$R$ </tex-math></inline-formula>. Specifically, we first introduce a new combinatorial structure called non-half-sum disjoint packing (NHSDP) which can be used to generate a coded caching scheme with <inline-formula> <tex-math>$K=O(F)$ </tex-math></inline-formula>. A class of new schemes is then obtained by constructing NHSDP. Theoretical analysis and numerical results demonstrate that 1) in comparison to existing schemes with linear subpacketization, the proposed scheme achieves a lower load; 2) proposed scheme also attains a lower load than some existing schemes with polynomial subpacketization in some cases; and 3) proposed scheme achieves load values comparable to those of existing schemes with exponential subpacketization in some cases. Furthermore, the newly introduced concept of NHSDP is closely related to classical combinatorial structures, including cyclic difference packings (CDP), non-three-term arithmetic progressions (NTAP), and perfect hash families (PHF). These relationships underscore the significance of NHSDP as a combinatorial structure of independent interest in the field of combinatorial design, even beyond its application to coded caching.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3753-3770"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148508893","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}

{"title":"A Generic Construction of q-ary Near-MDS Codes Supporting 2-Designs With Lengths Beyond q + 1","authors":"Hengfeng Liu;Chunming Tang;Zhengchun Zhou;Dongchun Han;Hao Chen","doi":"10.1109/TIT.2026.3681279","DOIUrl":"https://doi.org/10.1109/TIT.2026.3681279","url":null,"abstract":"A linear code with parameters <inline-formula> <tex-math>$[n, k, n {-} k + 1]$ </tex-math></inline-formula> is called maximum distance separable (MDS), and one with parameters <inline-formula> <tex-math>$[n, k, n - k]$ </tex-math></inline-formula> is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial <inline-formula> <tex-math>$t$ </tex-math></inline-formula>-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for <inline-formula> <tex-math>$t gt 2$ </tex-math></inline-formula>, all with length at most <inline-formula> <tex-math>$q + 1$ </tex-math></inline-formula>. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than <inline-formula> <tex-math>$q + 1$ </tex-math></inline-formula> are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first <italic>generic construction</i> of <inline-formula> <tex-math>$q$ </tex-math></inline-formula>-ary NMDS codes supporting 2-designs with lengths <italic>exceeding</i> <inline-formula> <tex-math>$q + 1$ </tex-math></inline-formula>. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3899-3909"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148510079","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}

{"title":"New Constructions of Binary Cyclic Codes With Both Relatively Large Minimum Distance and Dual Distance","authors":"Lingqi Zheng;Weijun Fang;Rongxing Qiu","doi":"10.1109/TIT.2026.3681593","DOIUrl":"https://doi.org/10.1109/TIT.2026.3681593","url":null,"abstract":"Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance <inline-formula> <tex-math>$d$ </tex-math></inline-formula> and dual distance <inline-formula> <tex-math>$d^{perp }$ </tex-math></inline-formula>. In recent years, much research has been devoted to improving the lower bound on <inline-formula> <tex-math>$d$ </tex-math></inline-formula>, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with length <inline-formula> <tex-math>$n=2^{m}-1$ </tex-math></inline-formula>, dimension near <inline-formula> <tex-math>$n/2$ </tex-math></inline-formula> and both relatively large minimum distance and dual distance. When <inline-formula> <tex-math>$m$ </tex-math></inline-formula> is even, we construct a family of binary cyclic codes with parameters <inline-formula> <tex-math>$[2^{m}-1,2^{m-1}pm 1,d]$ </tex-math></inline-formula>, where <inline-formula> <tex-math>$dge 2^{m/2}-1$ </tex-math></inline-formula> and <inline-formula> <tex-math>$d^{perp } ge 2^{m/2}$ </tex-math></inline-formula>. Both the minimum distance and the dual distance are significantly better than the previous results. When <inline-formula> <tex-math>$m$ </tex-math></inline-formula> is the product of two distinct primes, we construct some cyclic codes with dimensions <inline-formula> <tex-math>$k=(n+1)/2$ </tex-math></inline-formula> and <inline-formula> <tex-math>$dgt frac {n}{log _{2}n}$ </tex-math></inline-formula>, where the lower bound on the minimum distance is much larger than the square-root bound. When <inline-formula> <tex-math>$m$ </tex-math></inline-formula> is odd, we present two families of binary <inline-formula> <tex-math>$[2^{m}-1,2^{m-1},d]$ </tex-math></inline-formula> cyclic codes with <inline-formula> <tex-math>$dge 2^{(m+1)/2}-1$ </tex-math></inline-formula>, <inline-formula> <tex-math>$d^{perp } ge 2^{(m+1)/2}$ </tex-math></inline-formula> and <inline-formula> <tex-math>$dge 2^{(m+3)/2}-15$ </tex-math></inline-formula>, <inline-formula> <tex-math>$d^{perp } ge 2^{(m-1)/2}$ </tex-math></inline-formula> respectively, which leads that <inline-formula> <tex-math>$dcdot d^{perp } $ </tex-math></inline-formula> can reach <inline-formula> <tex-math>$2n$ </tex-math></inline-formula> asymptotically. To the best of our knowledge, for the binary cyclic codes with length <inline-formula> <tex-math>$n=2^{m}-1$ </tex-math></inline-formula> and dimension <inline-formula> <tex-math>$k=(npm 1)/2$ </tex-math></inline-formula>, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3716-3732"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511580","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}

{"title":"Low-Complexity Algebraic Soft Decoding of Hermitian Codes With Re-Encoding Transform","authors":"Jiwei Liang;Li Chen","doi":"10.1109/TIT.2026.3678324","DOIUrl":"https://doi.org/10.1109/TIT.2026.3678324","url":null,"abstract":"Algebraic codes are important in communication and storage systems where soft information is limited and decoding efficiency is critical. Among them, algebraic-geometric (AG) codes are promising candidates to replace the widely used Reed-Solomon (RS) codes. However, their more sophisticated algebraic structure results in high decoding complexity, hindering their practical application. This paper proposes a novel interpolation-based algebraic soft decoding (ASD) for one of the most important AG codes - Hermitian codes. Given a designed maximum decoding output list size (OLS), the interpolation module is constructed, and the decoding is formulated as finding a desired Gröbner basis of this module that contains the interpolation polynomial. The re-encoding transform (ReT) specifically designed for ASD is proposed, enabling the modified basis reduction (BR) interpolation to achieve lower complexity. The ReT is formulated by defining Lagrange interpolation polynomials over the Hermitian function fields and appropriately choosing the re-encoding points. By exploiting the linear code property, the ReT transforms a subset of interpolation points to have zero <inline-formula> <tex-math>$z$ </tex-math></inline-formula>-coordinates, which allows a common factor to be extracted from the module basis polynomials, thereby simplifying the interpolation. This paper further proposes an improved ReT employing more re-encoding points. It enables the common factor to have a greater degree, yielding a more significant complexity reduction. This also leads to an early termination for the decoding. Theoretical analysis and numerical results demonstrate that the two proposed ReT schemes efficiently facilitate the ASD for Hermitian codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3771-3783"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509112","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}