IEEE Transactions on Information Theory最新文献

{"title":"Relative Hulls and Their Variations of Narrow-Sense Primitive BCH Codes","authors":"Chunyu Gan;Chengju Li;Sihem Mesnager","doi":"10.1109/TIT.2026.3683768","DOIUrl":"https://doi.org/10.1109/TIT.2026.3683768","url":null,"abstract":"The relative hull of a linear code <inline-formula> <tex-math>${mathcal {C}}_{1}$ </tex-math></inline-formula> with respect to another linear code <inline-formula> <tex-math>${mathcal {C}}_{2}$ </tex-math></inline-formula> is defined as the intersection of <inline-formula> <tex-math>${mathcal {C}}_{1}$ </tex-math></inline-formula> and the dual of <inline-formula> <tex-math>${mathcal {C}}_{2}$ </tex-math></inline-formula>, i.e., <inline-formula> <tex-math>${mathcal {C}}_{1} cap {mathcal {C}}_{2}^{perp }$ </tex-math></inline-formula>. Anderson et al. demonstrated that the dimension of the relative hull can be adjusted, either repeatedly increased or decreased by one, until a certain bound is reached by substituting either <inline-formula> <tex-math>${mathcal {C}}_{1}$ </tex-math></inline-formula> or <inline-formula> <tex-math>${mathcal {C}}_{2}$ </tex-math></inline-formula> with its equivalent code. As a special class of linear codes, BCH codes are both theoretically significant and practically valuable for communication and storage systems due to their good algebraic structures and flexible error-correcting capabilities. This raises an important question: how does the dimension of the relative hull change when <inline-formula> <tex-math>${mathcal {C}}_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math>${mathcal {C}}_{2}$ </tex-math></inline-formula> are BCH codes or their equivalent codes? This paper focuses on the relative hull dimensions of narrow-sense primitive BCH codes, building on the insights from Anderson et al. (2024). We present several sufficient and necessary conditions based on the designed distances of BCH codes, which ensure that the dimensions of the relative hulls reach the lower or upper bounds for linear codes. Additionally, we study the parameters of the relative hulls of various classes of BCH codes, providing details on their dimensions and developing lower bounds for their minimum distances. Furthermore, we investigate how the dimensions of the relative hulls are affected when the BCH codes <inline-formula> <tex-math>${mathcal {C}}_{2}$ </tex-math></inline-formula> are replaced by their equivalent codes <inline-formula> <tex-math>${mathcal {C}}_{2}'$ </tex-math></inline-formula>.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3693-3715"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509778","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":"Doeblin Curves","authors":"Dongmin Lee;William Lu;Anuran Makur;Japneet Singh","doi":"10.1109/TIT.2026.3678229","DOIUrl":"https://doi.org/10.1109/TIT.2026.3678229","url":null,"abstract":"Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a <italic>Doeblin curve</i>—a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3556-3596"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511054","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":"Secondary Constructions of Plateaued Boolean Functions Through Addition of Indicators","authors":"Enes Pasalic;Sadmir Kudin;Samir Hodžić;Dilawar Abbas Khan","doi":"10.1109/TIT.2026.3685098","DOIUrl":"https://doi.org/10.1109/TIT.2026.3685098","url":null,"abstract":"Recently, the design of &lt;inline-formula&gt; &lt;tex-math&gt;$s$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-plateaued functions over &lt;inline-formula&gt; &lt;tex-math&gt;$ {mathbb {F}}_{2}^{n}$ &lt;/tex-math&gt;&lt;/inline-formula&gt;, also known as 3-valued Walsh spectra functions (taking the values from the set &lt;inline-formula&gt; &lt;tex-math&gt;$left {{{0, pm 2^{ frac {n+s}{2}}}}right }$ &lt;/tex-math&gt;&lt;/inline-formula&gt;), has been considered using an algorithmic approach (for example, IEEE Trans. Inf. Theory, vol. 70, no. 2, pp. 1408–1421, 2024) for the purpose of specifying balanced plateaued functions of maximal algebraic degree (called optimal). In this article, we identify these optimal plateaued functions within the Generalized Maiorana-McFarland (&lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {GMM}$ &lt;/tex-math&gt;&lt;/inline-formula&gt;) class and provide sufficient conditions that these functions do not admit nonzero linear structures. Moreover, we apply a similar technique to the one C. Carlet used for modifying bent functions in the Maiorana-McFarland (&lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {M}$ &lt;/tex-math&gt;&lt;/inline-formula&gt;) class (deriving the so-called &lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {C}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {D}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; classes), and we obtain new classes of plateaued functions from the &lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {GMM}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; class. However, when the function &lt;inline-formula&gt; &lt;tex-math&gt;$phi $ &lt;/tex-math&gt;&lt;/inline-formula&gt; in the definition of &lt;inline-formula&gt; &lt;tex-math&gt;$f(x,y)=x cdot phi (y) +delta _{0}(x)$ &lt;/tex-math&gt;&lt;/inline-formula&gt; is injective, then we cannot derive the subclass &lt;inline-formula&gt; &lt;tex-math&gt;${mathcal {D}}_{0}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; as in the case of bent functions. We show that this is still possible when &lt;inline-formula&gt; &lt;tex-math&gt;$phi $ &lt;/tex-math&gt;&lt;/inline-formula&gt; is not injective even though the sufficient conditions become more complicated compared to the bent case. Moreover, to ensure plateauedness within the &lt;inline-formula&gt; &lt;tex-math&gt;$mathcal {C}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; class, we were forced to impose certain conditions on the dual of a plateaued function, which is harder to handle compared to the bent case since the dual is defined on a subset/subspace of the ambient space. In general, using indicator functions of the form &lt;inline-formula&gt; &lt;tex-math&gt;$1_{E_{1}}(x)1_{E_{2}}(y)$ &lt;/tex-math&gt;&lt;/inline-formula&gt;, where the subspaces &lt;inline-formula&gt; &lt;tex-math&gt;$E_{1} subseteq {mathbb {F}} _{2}^{k}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt; &lt;tex-math&gt;$E_{2} subseteq {mathbb {F}} _{2}^{n-k}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; are suitably chosen, we show that it is possible to modify the function &lt;inline-formula&gt; &lt;tex-math&gt;$g(x,y)=x cdot phi (y) in {mathcal {GMM}}_{k}^{n}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; while preserving plateauedness, so that &lt;inline-formula&gt; &lt;tex-math&gt;$f(x,y)=x cdot phi (y) + 1_{E_{1}}(x)1_{E_{2}}(y)$ &lt;/tex-math&gt;&lt;/inline-formula&gt; is outside the &lt;inline-formula","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"4329-4340"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148508912","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":"Instance-Dependent Uniform Tail Bounds for Empirical Processes","authors":"Sohail Bahmani","doi":"10.1109/TIT.2026.3677108","DOIUrl":"https://doi.org/10.1109/TIT.2026.3677108","url":null,"abstract":"We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial “deflation” step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the “deflated function class” in terms of a generalization of Talagrand’s <inline-formula> <tex-math>$gamma $ </tex-math></inline-formula> functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cramér functions. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their “incomplete” versions under suitable moment conditions.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"4187-4202"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509631","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 and Lower Bounds for Maximally Recoverable Grid Codes","authors":"Joshua Brakensiek;Manik Dhar;Sivakanth Gopi","doi":"10.1109/TIT.2026.3685533","DOIUrl":"https://doi.org/10.1109/TIT.2026.3685533","url":null,"abstract":"In this paper, we continue the study of Maximally Recoverable (MR) Grid Codes initiated by Gopalan et al. [SODA 2017]. More precisely, we study codes over an <inline-formula> <tex-math>$m times n$ </tex-math></inline-formula> grid topology with one parity check per row and column of the grid along with <inline-formula> <tex-math>$h ge 1$ </tex-math></inline-formula> global parity checks. Previous works have largely focused on the setting in which <inline-formula> <tex-math>$m = n$ </tex-math></inline-formula>, where explicit constructions require field size which is exponential in <inline-formula> <tex-math>$n$ </tex-math></inline-formula>. Motivated by practical applications, we consider the regime in which <inline-formula> <tex-math>$m,h$ </tex-math></inline-formula> are constants and <inline-formula> <tex-math>$n$ </tex-math></inline-formula> is growing. In this setting, we provide a number of new explicit constructions whose field size is polynomial in <inline-formula> <tex-math>$n$ </tex-math></inline-formula>. We further complement these results with new field size lower bounds.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3820-3829"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509728","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":"Asymptotic Analysis of Nonlinear One-Bit Precoding in Massive MIMO Systems via Approximate Message Passing","authors":"Zheyu Wu;Junjie Ma;Ya-Feng Liu;Bruno Clerckx","doi":"10.1109/TIT.2026.3682569","DOIUrl":"https://doi.org/10.1109/TIT.2026.3682569","url":null,"abstract":"Massive multiple-input multiple-output (MIMO) systems employing one-bit digital-to-analog converters offer a hardware-efficient solution for wireless communications. However, the one-bit constraint poses significant challenges for precoding design, as it transforms the problem into a discrete and nonconvex optimization task. In this paper, we investigate a widely adopted “convex-relaxation-then-quantization” approach for nonlinear symbol-level one-bit precoding. Specifically, we first solve a convex relaxation of the discrete minimum mean square error precoding problem, and then quantize the solution to satisfy the one-bit constraint. Focusing on a real-valued system with an independently and identically distributed (i.i.d.) Gaussian channel, we develop a novel analytical framework based on approximate message passing (AMP) to characterize the high-dimensional asymptotic performance of the considered scheme. The key technical ingredient is an auxiliary AMP iteration that dedicatedly incorporates the nonlinear quantization function into the state evolution analysis. With the proposed framework, we derive a closed-form expression for the symbol error probability (SEP) at the receiver side in the large-system limit, which provides a quantitative characterization of how model and system parameters affect the SEP performance. Our empirical results suggest that the <inline-formula> <tex-math>$ell _{infty } ^{2}$ </tex-math></inline-formula> regularizer, when paired with an optimally chosen regularization parameter, achieves optimal SEP performance within a broad class of convex regularization functions. As a first step towards a theoretical justification, we prove the optimality of the <inline-formula> <tex-math>$ell _{infty } ^{2}$ </tex-math></inline-formula> regularizer within the mixed <inline-formula> <tex-math>$ell _{infty } ^{2}$ </tex-math></inline-formula>-<inline-formula> <tex-math>$ell _{2}^{2}$ </tex-math></inline-formula> regularization functions.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3950-3978"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148509775","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":"Universal Framework for Parametric Constrained Coding","authors":"Adir Kobovich;Orian Leitersdorf;Daniella Bar-Lev;Eitan Yaakobi","doi":"10.1109/TIT.2026.3672710","DOIUrl":"https://doi.org/10.1109/TIT.2026.3672710","url":null,"abstract":"Constrained coding is a subfield of coding theory that tackles efficient communication under constraints. While fixed constraints (e.g., a fixed set of substrings may not appear in transmitted messages) have a general optimal solution, there is increasing demand for supporting <italic>parametric</i> constraints that are dependent on the message length and portray some property (e.g., no <inline-formula> <tex-math>$log (n)$ </tex-math></inline-formula> consecutive zeros). Several works have tackled such parametric constraints through <italic>iterative</i> algorithms, yet they require complex constructions specific to each constraint to guarantee convergence through <italic>monotonic progression</i>. In this paper, we propose a universal framework for tackling <italic>any</i> parametric constraint problem through a new <italic>simple</i> iterative algorithm. By reducing an execution of this iterative algorithm to an acyclic graph traversal, we prove a surprising result that guarantees convergence with low average time complexity <italic>even without requiring any monotonic progression</i>. We demonstrate the effectiveness of this universal framework, with much of our focus on the special case of single-symbol redundancy, while also considering a variety of both <italic>local</i> and <italic>global</i> constraints. We begin by exploring the local constraints involving illegal substrings of variable length, where the construction essentially iteratively replaces forbidden windows. This local algorithm is applied to various fundamental constraints, achieving state-of-the-art results through simple adaptations of the universal algorithm. We then continue by exploring global constraints, and demonstrate the effectiveness of the proposed construction on repeat-free encoding, reverse-complement encoding and DNA data storage. Overall, the proposed framework generates state-of-the-art constructions with significant ease while also enabling the simultaneous integration of multiple constraints for the first time.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3678-3692"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511062","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":"Smoothed Analysis in Compressed Sensing","authors":"Elad Aigner-Horev;Dan Hefetz;Michael Trushkin","doi":"10.1109/TIT.2026.3681197","DOIUrl":"https://doi.org/10.1109/TIT.2026.3681197","url":null,"abstract":"Arbitrary matrices <inline-formula> <tex-math>$M in mathsf {R}^{m times n}$ </tex-math></inline-formula>, randomly perturbed in an additive manner using a random matrix <inline-formula> <tex-math>$R in mathsf {R}^{m times n}$ </tex-math></inline-formula>, are shown to asymptotically almost surely satisfy the so-called <italic>robust null space property</i>. Whilst insisting on an asymptotically optimal order of magnitude for <inline-formula> <tex-math>$m$ </tex-math></inline-formula> required to attain <italic>unique reconstruction</i> via <inline-formula> <tex-math>$ell _{1}$ </tex-math></inline-formula>-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix <inline-formula> <tex-math>$M$ </tex-math></inline-formula> as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix <inline-formula> <tex-math>$R$ </tex-math></inline-formula>. Starting with sub-gaussian entries for <inline-formula> <tex-math>$R$ </tex-math></inline-formula>, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for <inline-formula> <tex-math>$M$ </tex-math></inline-formula>; the first is <inline-formula> <tex-math>$|M|_{infty } $ </tex-math></inline-formula> and the second is a localised notion of the Frobenius norm of <inline-formula> <tex-math>$M$ </tex-math></inline-formula> (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is <italic>Mendelson’s small-ball method</i> (<italic>Learning without concentration</i>, J. ACM, Vol. 62, 2015).","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"4399-4414"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511117","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":"Canonical Form and Finite Blocklength Bounds for Stabilizer Codes","authors":"Dimiter Ostrev","doi":"10.1109/TIT.2026.3688060","DOIUrl":"https://doi.org/10.1109/TIT.2026.3688060","url":null,"abstract":"First, a canonical form for stabilizer parity check matrices of arbitrary size and rank is derived. Next, it is shown that the closely related canonical form of the Clifford group can be computed in time <inline-formula> <tex-math>$O(n^{3})$ </tex-math></inline-formula> for <inline-formula> <tex-math>$n$ </tex-math></inline-formula> qubits, which improves upon the previously known time <inline-formula> <tex-math>$O(n^{6})$ </tex-math></inline-formula>. Finally, the related problem of finite blocklength bounds for stabilizer codes and Pauli noise is studied. A finite blocklength refinement of the hashing bound is derived, and it is shown that no argument that uses guessing the error as a substitute for guessing the coset can lead to a significantly better achievability bound.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3979-4000"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511122","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":"On Higher-Order Minimal Linear Codes","authors":"Changsu Wang;Jie Peng","doi":"10.1109/TIT.2026.3681362","DOIUrl":"https://doi.org/10.1109/TIT.2026.3681362","url":null,"abstract":"Secret sharing was introduced by Shamir to distribute a secret among a group of participants. Later, Massey generalized this approach using minimal linear codes. Recent research has extensively explored the basic properties, code parameters, and constructions of minimal linear codes from algebraic, combinatorial, and geometric perspectives. In this paper, we further generalize Massey’s scheme to multi-secret sharing by introducing <italic>higher-order minimal linear codes</i>. We present some fundamental properties of these codes, including two necessary and sufficient conditions for <inline-formula> <tex-math>$r$ </tex-math></inline-formula>th-order minimality. These conditions generalize the Heng–Ding–Zhou and the Lu–Wu–Cao conditions for ordinary minimal linear codes, respectively. We also derive bounds on the parameters of higher-order minimal linear codes. Importantly, we prove that these codes are asymptotically good, exhibiting both non-vanishing rate and relative distance as the code length tends to infinity. A geometric characterization establishes an equivalence between <inline-formula> <tex-math>$r$ </tex-math></inline-formula>th-order minimal linear codes and cutting <inline-formula> <tex-math>$r$ </tex-math></inline-formula>-blocking sets in projective geometry. Finally, we provide higher-order minimal linear codes using punctured simplex codes, Kronecker products and subfield subcodes, respectively.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"72 6","pages":"3799-3808"},"PeriodicalIF":2.6,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148511584","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}