IEEE Transactions on Information Theory最新文献

{"title":"Random Staircase Generator Matrix Codes: Coding Theorem, Performance Analysis, and Code Design","authors":"Qianfan Wang;Yiwen Wang;Yixin Wang;Jifan Liang;Xiao Ma","doi":"10.1109/TIT.2025.3541734","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541734","url":null,"abstract":"In this paper, we present a class of codes, referred to as random staircase generator matrix codes (SGMCs), which have staircase-like generator matrices. In the infinite-length region, we prove that the random SGMC is capacity-achieving over binary-input output-symmetric (BIOS) channels. In the finite-length region, we propose the generalized representative ordered statistics decoding with local constraints (LC-ROSD) algorithm for the SGMCs. The most distinguished feature of the SGMCs with LC-ROSD is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE), avoiding the serial GE of conventional OSD and supporting a potential low decoding latency, as implied from simulations. To analyze the performance of random SGMCs in the finite-length region, we derive the ensemble weight spectrum and invoke the conventional union bound. We also derive a partially random coding union (RCU) bound, which is tighter than the conventional one and is used as a criterion to design the SGMCs. Staircase-like generator matrices allow us to derive a series of (tighter and tighter) lower bounds based on the second-order Bonferroni inequality with the incremental number of codewords. The numerical results show that the decoding performance can match well with the proposed partially RCU bound for different code rates and different profiles. The numerical results also show that the tailored SGMCs with the LC-ROSD algorithm can approach the finite-length performance bound, outperforming the 5G low-density parity-check (LDPC) codes, 5G polar codes, and Reed-Muller (RM) codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3497-3509"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143875154","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":"Variable-Length Feedback Codes Over Known and Unknown Channels With Non-Vanishing Error Probabilities","authors":"Recep Can Yavas;Vincent Y. F. Tan","doi":"10.1109/TIT.2025.3541629","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541629","url":null,"abstract":"We study variable-length feedback (VLF) codes with noiseless feedback for discrete memoryless channels. We present a novel non-asymptotic bound, which analyzes the average error probability and average decoding time of our modified Yamamoto-Itoh scheme. We then optimize the parameters of our code in the asymptotic regime where the average error probability <inline-formula> <tex-math>$epsilon $ </tex-math></inline-formula> remains a constant as the average decoding time <italic>N</i> approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.’s (2011) achievability bound. We also develop a universal VLF code that does not rely on the knowledge of the underlying channel parameters. Our universal VLF code employs the empirical mutual information as its decoding metric and universalizes the code by Polyanskiy et al. (2011). We derive a second-order achievability bound for universal VLF codes. Our results for both VLF and universal VLF codes are extended to the additive white Gaussian noise channel with an average power constraint. The former yields an improvement over Truong and Tan’s (2017) achievability bound. The proof of our results for universal VLF codes uses a refined version of the method of types and an asymptotic expansion from the nonlinear renewal theory literature.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3271-3286"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873126","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 Comprehensive Study on Ziv-Zakai Lower Bounds on the MMSE","authors":"Minoh Jeong;Alex Dytso;Martina Cardone","doi":"10.1109/TIT.2025.3541987","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541987","url":null,"abstract":"This paper explores Bayesian lower bounds on the minimum mean squared error (MMSE) that belong to the well-known Ziv-Zakai family. The Ziv-Zakai technique relies on connecting the bound to an <inline-formula> <tex-math>$mathsf M$ </tex-math></inline-formula>-ary hypothesis testing problem. There are three versions of the Ziv-Zakai bound (ZZB): the first version relies on the so-called <italic>valley-filling function</i>, the second one is a relaxation of the first bound which omits the valley-filling function, and the third one, namely the single-point ZZB (SZZB), replaces the integration present in the first two bounds with a single point maximization. The first part of this paper focuses on providing the most general version of the bounds. It is shown that these bounds hold without any assumption on the distribution of the estimand. This makes the bounds applicable to discrete and mixed distributions. Then, the SZZB is extended to an <inline-formula> <tex-math>$mathsf M$ </tex-math></inline-formula>-ary setting and a version of it that holds for the multivariate setting is provided. In the second part, general properties of these bounds are provided. First, unlike the Bayesian <italic>Cramér-Rao bound</i>, it is shown that all the versions of the ZZB <italic>tensorize</i>. Second, a characterization of the <italic>high-noise</i> asymptotic is provided, which is used to argue about the tightness of the bounds. Third, a complete <italic>low-noise</i> asymptotic is provided under the assumptions of mixed-input distributions and Gaussian additive noise channels. In the low-noise, it is shown that the ZZB is generally tight, but there are examples for which the SZZB is not tight. In the third part, the tightness of the bounds is evaluated. First, it is shown that in the low-noise regime the ZZB without the valley-filling function, and, therefore, also the ZZB with the valley-filling function, are tight for mixed-input distributions and Gaussian additive noise channels. Second, for discrete inputs it is shown that the ZZB with the valley-filling function is always sub-optimal, and equal to zero without the valley-filling function. Third, unlike for the ZZB, an example is shown for which the SZZB is tight to the MMSE for discrete inputs. Fourth, sufficient and necessary conditions for the tightness of the bounds are provided. Finally, some examples are provided in which the bounds in the Ziv-Zakai family outperform other well-known Bayesian lower bounds, namely the Cramér-Rao bound and the maximum entropy bound.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 4","pages":"3214-3236"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667259","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":"Blind Interference Alignment for MapReduce: Exploiting Side-Information With Reconfigurable Antennas","authors":"Yuxiang Lu;Syed A. Jafar","doi":"10.1109/TIT.2025.3541808","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541808","url":null,"abstract":"In order to explore how blind interference alignment (BIA) schemes may take advantage of side-information in computation tasks, we study the degrees of freedom (DoF) of a <italic>K</i> user wireless network setting that arises in full-duplex wireless MapReduce applications. In this setting the receivers are assumed to have reconfigurable antennas and channel knowledge, while the transmitters have neither, i.e., the transmitters lack channel knowledge and are only equipped with conventional antennas. The central ingredient of the problem formulation is the message structure arising out of the Shuffle phase of MapReduce, whereby each transmitter has a subset of messages that need to be delivered to various receivers, and each receiver has a subset of messages available to it in advance as side-information. We approach this problem by decomposing it into distinctive stages that help identify key ingredients of the overall solution. The novel elements that emerge from the first stage, called broadcast with groupcast messages, include an outer maximum distance separable (MDS) code structure at the transmitter, and an algorithm for iteratively determining groupcast-optimal reconfigurable antenna switching patterns at the receiver to achieve intra-message (among the symbols of the same message) alignment. The next stage, called unicast with side-information, reveals optimal inter-message (among symbols of different messages) alignment patterns to exploit side-information, and by a relabeling of messages, connects to the desired MapReduce setting.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 4","pages":"2604-2625"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143676084","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":"Joint Message Detection and Channel Estimation for Unsourced Random Access in Cell-Free User-Centric Wireless Networks","authors":"Burak Çakmak;Eleni Gkiouzepi;Manfred Opper;Giuseppe Caire","doi":"10.1109/TIT.2025.3541281","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541281","url":null,"abstract":"We consider unsourced random access (uRA) in a cell-free (CF) user-centric wireless network, where a large number of potential users compete for a random access slot, while only a finite subset is active. The random access users transmit codewords of length <italic>L</i> symbols from a shared codebook, which are received by <italic>B</i> geographically distributed radio units (RUs), each equipped with <italic>M</i> antennas. Our goal is to devise and analyze a <italic>centralized</i> decoder to detect the transmitted messages (without prior knowledge of the active users) and estimate the corresponding channel state information. A specific challenge lies in the fact that, due to the geographically distributed nature of the CF network, there is no fixed correspondence between codewords and large-scale fading coefficients (LSFCs). This makes current activity detection approaches which make use of this fixed LSFC-codeword association not directly applicable. To overcome this problem, we propose a scheme where the access codebook is partitioned in location-based subcodes, such that users in a particular location make use of the corresponding subcode. The joint message detection and channel estimation is obtained via a novel <italic>Approximated Message Passing</i> (AMP) algorithm for a linear superposition of matrix-valued sources corrupted by noise. The statistical asymmetry in the fading profile and message activity leads to <italic>different statistics</i> for the matrix sources, which distinguishes the AMP formulation from previous cases. In the regime where the codebook size scales linearly with <italic>L</i>, while <italic>B</i> and <italic>M</i> are fixed, we present a rigorous high-dimensional (but finite-sample) analysis of the proposed AMP algorithm. Exploiting this, we then present a precise (and rigorous) large-system analysis of the message missed-detection and false-alarm rates, as well as the channel estimation mean-square error. The resulting system allows the seamless formation of user-centric clusters and very low latency beamformed uplink-downlink communication without explicit user-RU association, pilot allocation, and power control. This makes the proposed scheme highly appealing for low-latency random access communications in CF networks.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3614-3643"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870973","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":"Covering Radii and Deep Holes of Two Classes of Extended Twisted GRS Codes and Their Applications","authors":"Yang Li;Shixin Zhu;Zhonghua Sun","doi":"10.1109/TIT.2025.3541799","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541799","url":null,"abstract":"Maximum distance separable (MDS) codes that are not monomially equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes, which have important applications in communication and cryptography. Covering radii and deep holes of linear codes are closely related to their decoding problems. In the literature, the covering radii and deep holes of GRS codes have been extensively studied, while little is known about non-GRS MDS codes. In this paper, we study two classes of extended twisted generalized Reed-Solomon (ETGRS) codes involving their non-GRS MDS properties, covering radii, and deep holes. In other words, we obtain two classes of non-GRS MDS codes with known covering radii and deep holes. As applications, we further directly derive more non-GRS MDS codes, and get some results on the existence of their error-correcting pairs. As a byproduct, we find some connections between the well-known Roth-Lempel codes and these two classes ETGRS codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3516-3530"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143875183","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":"MDS Variable Generation and Secure Summation With User Selection","authors":"Yizhou Zhao;Hua Sun","doi":"10.1109/TIT.2025.3541551","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541551","url":null,"abstract":"A collection of <italic>K</i> random variables are called <inline-formula> <tex-math>$(K,n)$ </tex-math></inline-formula>-MDS if any <italic>n</i> of the <italic>K</i> variables are independent and determine all remaining variables. In the MDS variable generation problem, <italic>K</i> users wish to generate variables that are <inline-formula> <tex-math>$(K,n)$ </tex-math></inline-formula>-MDS using a randomness variable owned by each user. We show that to generate 1 bit of <inline-formula> <tex-math>$(K,n)$ </tex-math></inline-formula>-MDS variables for each <inline-formula> <tex-math>$n in {1,2,cdots , K}$ </tex-math></inline-formula>, the minimum size of the randomness variable at each user is <inline-formula> <tex-math>$1 + 1/2 + cdots + 1/K$ </tex-math></inline-formula> bits. An intimately related problem is secure summation with user selection, where a server may select an arbitrary subset of <italic>K</i> users and securely compute the sum of the inputs of the selected users. We show that to compute 1 bit of an arbitrarily chosen sum securely, the minimum size of the key held by each user is <inline-formula> <tex-math>$1 + 1/2 + cdots + 1/(K-1)$ </tex-math></inline-formula> bits, whose achievability uses the generation of <inline-formula> <tex-math>$(K,n)$ </tex-math></inline-formula>-MDS variables for <inline-formula> <tex-math>$n in {1,2,cdots ,K-1}$ </tex-math></inline-formula>.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 4","pages":"3129-3141"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667375","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 k-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction","authors":"Kuan Cheng;Elena Grigorescu;Xin Li;Madhu Sudan;Minshen Zhu","doi":"10.1109/TIT.2025.3541375","DOIUrl":"https://doi.org/10.1109/TIT.2025.3541375","url":null,"abstract":"The goal of the trace reconstruction problem is to recover a string &lt;inline-formula&gt; &lt;tex-math&gt;$mathbf {x}in {0,1}^{n}$ &lt;/tex-math&gt;&lt;/inline-formula&gt; given many independent &lt;italic&gt;traces&lt;/i&gt; of &lt;bold&gt;x&lt;/b&gt;, where a trace is a subsequence obtained from deleting bits of &lt;bold&gt;x&lt;/b&gt; independently with some given probability &lt;inline-formula&gt; &lt;tex-math&gt;$pin [0,1$ &lt;/tex-math&gt;&lt;/inline-formula&gt;). A recent result of Chase (STOC 2021) shows how &lt;bold&gt;x&lt;/b&gt; can be determined (in exponential time) from &lt;inline-formula&gt; &lt;tex-math&gt;$exp ({O}(n^{1/5})log ^{5} n)$ &lt;/tex-math&gt;&lt;/inline-formula&gt; traces. This is the state-of-the-art result on the sample complexity of trace reconstruction. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the bound of Chase, which is based on statistics of arbitrary length-&lt;italic&gt;k&lt;/i&gt; subsequences, can also be obtained by considering the “&lt;italic&gt;k&lt;/i&gt;-mer statistics”, i.e., statistics regarding occurrences of &lt;italic&gt;contiguous k&lt;/i&gt;-bit strings (a.k.a, &lt;italic&gt;k-mers&lt;/i&gt;) in the initial string &lt;bold&gt;x&lt;/b&gt;, for &lt;inline-formula&gt; &lt;tex-math&gt;$k = 2n^{1/5}$ &lt;/tex-math&gt;&lt;/inline-formula&gt;. Mazooji and Shomorony (arXiv.2210.10917) show that such statistics (called &lt;italic&gt;k&lt;/i&gt;-mer density map) can be estimated within &lt;inline-formula&gt; &lt;tex-math&gt;$varepsilon $ &lt;/tex-math&gt;&lt;/inline-formula&gt; accuracy from &lt;inline-formula&gt; &lt;tex-math&gt;$ {mathrm {poly}} (n, 2^{k}, 1/ {varepsilon })$ &lt;/tex-math&gt;&lt;/inline-formula&gt; traces. We call an algorithm to be &lt;italic&gt;k-mer-based&lt;/i&gt; if it reconstructs &lt;bold&gt;x&lt;/b&gt; given estimates of the &lt;italic&gt;k&lt;/i&gt;-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that any &lt;italic&gt;k&lt;/i&gt;-mer-based algorithm for trace reconstruction must use &lt;inline-formula&gt; &lt;tex-math&gt;$exp (Omega (n^{1/5} sqrt {log n}))$ &lt;/tex-math&gt;&lt;/inline-formula&gt; traces, thus establishing the optimality of this number of traces. The analysis of this result also shows that the analysis technique used by Chase (STOC 2021) is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. This result is shown by considering an appropriate class of real polynomials, that have been previously studied in the context of trace estimation (De, O’Donnell, Servedio. Annals of Probability 2019; Nazarov, Peres. STOC 2017), and proving that two of these polynomials are very close to each other on an arc in the complex plane. Our proof of the proximity of such polynomials uses new technical ingredients that allow us to focus on just a few coefficients of these polynomials. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorith","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 4","pages":"2591-2603"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143676099","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":"Non-Reed-Solomon Type Cyclic MDS Codes","authors":"Fagang Li;Yangyang Chen;Hao Chen;Yongfeng Niu","doi":"10.1109/TIT.2025.3538220","DOIUrl":"https://doi.org/10.1109/TIT.2025.3538220","url":null,"abstract":"As cyclic codes and maximum distance separable (MDS) codes, cyclic MDS codes have very nice structures and properties, which have been intensively investigated in literature due to their theoretical interest and practical importance. Particularly, abundant cyclic MDS codes have been determined and constructed for many parameters and most of them were proved to be equivalent to generalized Reed-Solomon (GRS) codes. Hence it is a challenging task to construct non-Reed-Solomon type cyclic MDS codes. In this work, we obtain many new cyclic MDS codes for certain parameters by determining the solutions of the system of polynomial equations. Moreover, by determining the dimension of the Schur square of an MDS code, we can easily show that all of our constructed codes are not equivalent to GRS codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 5","pages":"3489-3496"},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143875157","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 Bounds for Generalized Column Distances and Construction of Convolutional Codes","authors":"Xu Pan;Hao Chen;Chunming Tang;Xueyan Chen","doi":"10.1109/TIT.2025.3539914","DOIUrl":"https://doi.org/10.1109/TIT.2025.3539914","url":null,"abstract":"Based on known bounds for relative generalized Hamming weights of linear codes, we provide several new bounds for generalized column distances of convolutional codes, including the Griesmer-type bound for generalized column distances. Then we construct several infinite families of convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is small compared with the length of these convolutional codes by using cyclic codes, negacyclic codes and GRS codes. In particular, we obtain some convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is zero or one. Next we prove that the 2-generalized column distance sequence <inline-formula> <tex-math>${d_{2,j}(mathcal {C})}_{j=1}^{infty }$ </tex-math></inline-formula> of any convolutional code <inline-formula> <tex-math>$mathcal {C}$ </tex-math></inline-formula> is increasing and bounded from above, and the limit of the sequence <inline-formula> <tex-math>${d_{2,j}(mathcal {C})}_{j=1}^{infty }$ </tex-math></inline-formula> is related to the 2-generalized Hamming weight of the convolutional code <inline-formula> <tex-math>$mathcal {C}$ </tex-math></inline-formula>. For <inline-formula> <tex-math>$ige 3$ </tex-math></inline-formula>, we prove that the <italic>i</i>-generalized column distance sequence <inline-formula> <tex-math>${d_{i,j}(mathcal {C})}_{j=lceil frac {i}{k}-1rceil }^{infty }$ </tex-math></inline-formula> of any convolutional code <inline-formula> <tex-math>$mathcal {C}$ </tex-math></inline-formula> is bounded above and below.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 4","pages":"2576-2590"},"PeriodicalIF":2.2,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143698322","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}