Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. The required field size for an $[n,k]_{q^{m}}$ LRS codes with support-constrained generator matrix is $qgeq ell +1$ and $mgeq max _{lin [ell]}{k-1+log _{q}k, n_{l}}$ , where $ell $ is the number of blocks and $n_{l}$ is the size of the l-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.
{"title":"Linearized Reed-Solomon Codes With Support-Constrained Generator Matrix and Applications in Multi-Source Network Coding","authors":"Hedongliang Liu;Hengjia Wei;Antonia Wachter-Zeh;Moshe Schwartz","doi":"10.1109/TIT.2024.3503770","DOIUrl":"https://doi.org/10.1109/TIT.2024.3503770","url":null,"abstract":"Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. The required field size for an <inline-formula> <tex-math>$[n,k]_{q^{m}}$ </tex-math></inline-formula> LRS codes with support-constrained generator matrix is <inline-formula> <tex-math>$qgeq ell +1$ </tex-math></inline-formula> and <inline-formula> <tex-math>$mgeq max _{lin [ell]}{k-1+log _{q}k, n_{l}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math>$ell $ </tex-math></inline-formula> is the number of blocks and <inline-formula> <tex-math>$n_{l}$ </tex-math></inline-formula> is the size of the l-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 2","pages":"895-913"},"PeriodicalIF":2.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143106999","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 : 2024-11-27DOI: 10.1109/TIT.2024.3507278
Xiangrui Meng;Jian Gao;Qingxiang Cui;Fang-Wei Fu
Constantinescu et al. introduced the homogeneous weight on the integer residue ring $mathbb {Z}_{m}$ which can reflect more information compared with the Hamming weight. Few homogeneous weight linear codes over finite chain rings have important applications in cryptography, lattices, modular forms and combinatorics. In this paper, we construct an infinite class of cyclic codes over the finite chain ring $mathbb {F}_{p^{t}}[omega]/(omega ^{2})$ by the trace function, and determine their homogeneous weight distributions by applying the theory of exponential sums. In order to investigate the minimality of linear codes over finite chain rings, we firstly present the necessary and sufficient condition for linear codes over the finite chain ring $mathbb {F}_{p^{t}}[omega]/(omega ^{2})$ to be minimal or almost minimal by the Hamming weights of codewords. Then, based on the proposed condition and few Hamming weight cyclic codes, we give several classes of minimal and almost minimal linear codes. Furthermore, we derive several families of strongly regular graphs, strongly walk-regular graphs and triple sum sets by few homogeneous weight linear codes.
{"title":"Homogeneous Weight Distributions of Cyclic Codes Over Finite Chain Rings","authors":"Xiangrui Meng;Jian Gao;Qingxiang Cui;Fang-Wei Fu","doi":"10.1109/TIT.2024.3507278","DOIUrl":"https://doi.org/10.1109/TIT.2024.3507278","url":null,"abstract":"Constantinescu et al. introduced the homogeneous weight on the integer residue ring <inline-formula> <tex-math>$mathbb {Z}_{m}$ </tex-math></inline-formula> which can reflect more information compared with the Hamming weight. Few homogeneous weight linear codes over finite chain rings have important applications in cryptography, lattices, modular forms and combinatorics. In this paper, we construct an infinite class of cyclic codes over the finite chain ring <inline-formula> <tex-math>$mathbb {F}_{p^{t}}[omega]/(omega ^{2})$ </tex-math></inline-formula> by the trace function, and determine their homogeneous weight distributions by applying the theory of exponential sums. In order to investigate the minimality of linear codes over finite chain rings, we firstly present the necessary and sufficient condition for linear codes over the finite chain ring <inline-formula> <tex-math>$mathbb {F}_{p^{t}}[omega]/(omega ^{2})$ </tex-math></inline-formula> to be minimal or almost minimal by the Hamming weights of codewords. Then, based on the proposed condition and few Hamming weight cyclic codes, we give several classes of minimal and almost minimal linear codes. Furthermore, we derive several families of strongly regular graphs, strongly walk-regular graphs and triple sum sets by few homogeneous weight linear codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 2","pages":"955-974"},"PeriodicalIF":2.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143106957","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 : 2024-11-26DOI: 10.1109/TIT.2024.3506866
Eray Can Elumar;Cem Tekin;Osman Yağan
Multi-armed bandits is a sequential decision-making problem where an agent must choose between multiple actions to maximize its cumulative reward over time, while facing uncertainty about the rewards associated with each action. The challenge lies in balancing the exploration of potentially higher-rewarding actions with the exploitation of known high-reward actions. We consider a multi-armed bandit problem with probes, where before pulling an arm, the decision-maker is allowed to probe one of the K arms for a cost $cgeq 0$ to observe its reward. We introduce a new regret definition that is based on the expected reward of the optimal action. We develop UCBP, a novel algorithm that utilizes this strategy to achieve a gap-independent regret upper bound that scales with the number of rounds T as $ O(sqrt {KTlog T})$ , and an order optimal gap-dependent upper bound of $ O(Klog T)$ . As a baseline, we introduce UCB-naive-probe, a naive UCB-based approach which has a gap-independent regret upper bound of $O(Ksqrt {Tlog T})$ , and gap-dependent regret bound of $O(K^{2}log T)$ ; and TSP, the Thompson sampling version of UCBP. In empirical simulations, UCBP outperforms UCB-naive-probe, and performs similarly to TSP, verifying the utility of UCBP and TSP algorithms in practical settings.
{"title":"Multi-Armed Bandits With Costly Probes","authors":"Eray Can Elumar;Cem Tekin;Osman Yağan","doi":"10.1109/TIT.2024.3506866","DOIUrl":"https://doi.org/10.1109/TIT.2024.3506866","url":null,"abstract":"Multi-armed bandits is a sequential decision-making problem where an agent must choose between multiple actions to maximize its cumulative reward over time, while facing uncertainty about the rewards associated with each action. The challenge lies in balancing the exploration of potentially higher-rewarding actions with the exploitation of known high-reward actions. We consider a multi-armed bandit problem with probes, where before pulling an arm, the decision-maker is allowed to probe one of the K arms for a cost \u0000<inline-formula> <tex-math>$cgeq 0$ </tex-math></inline-formula>\u0000 to observe its reward. We introduce a new regret definition that is based on the expected reward of the optimal action. We develop UCBP, a novel algorithm that utilizes this strategy to achieve a gap-independent regret upper bound that scales with the number of rounds T as \u0000<inline-formula> <tex-math>$ O(sqrt {KTlog T})$ </tex-math></inline-formula>\u0000, and an order optimal gap-dependent upper bound of \u0000<inline-formula> <tex-math>$ O(Klog T)$ </tex-math></inline-formula>\u0000. As a baseline, we introduce UCB-naive-probe, a naive UCB-based approach which has a gap-independent regret upper bound of \u0000<inline-formula> <tex-math>$O(Ksqrt {Tlog T})$ </tex-math></inline-formula>\u0000, and gap-dependent regret bound of \u0000<inline-formula> <tex-math>$O(K^{2}log T)$ </tex-math></inline-formula>\u0000; and TSP, the Thompson sampling version of UCBP. In empirical simulations, UCBP outperforms UCB-naive-probe, and performs similarly to TSP, verifying the utility of UCBP and TSP algorithms in practical settings.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"618-643"},"PeriodicalIF":2.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890254","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}
Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.
{"title":"Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes","authors":"Martianus Frederic Ezerman;Markus Grassl;San Ling;Ferruh Özbudak;Buket Özkaya","doi":"10.1109/TIT.2024.3503420","DOIUrl":"https://doi.org/10.1109/TIT.2024.3503420","url":null,"abstract":"Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"499-517"},"PeriodicalIF":2.2,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890145","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 : 2024-11-25DOI: 10.1109/TIT.2024.3493733
{"title":"IEEE Transactions on Information Theory Publication Information","authors":"","doi":"10.1109/TIT.2024.3493733","DOIUrl":"https://doi.org/10.1109/TIT.2024.3493733","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"C2-C2"},"PeriodicalIF":2.2,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10767145","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142753905","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}
We present secondary constructions of vectorial functions respectively partitions of elementary abelian groups, which simultaneously yield vectorial dual-bent functions with certain properties, bent partitions, and under some conditions, Latin square partial difference set packings (LP-packings). First, we analyse constructions via the direct sum of vectorial functions and then present a version of the generalized Maiorana-McFarland construction. Next, we generalize a construction of vectorial dual-bent functions by Wang, Fu, and Wei (2023). Finally, we use a lifting procedure of LP-packings from Jedwab and Li (2021) to construct vectorial dual-bent functions, bent partitions, and LP-packings in elementary abelian groups. With these constructions, a large variety of vectorial bent functions, bent partitions, LP-packings, and related amorphic association schemes can be obtained.
本文给出了向量函数的二次构造，分别对初等阿贝尔群进行了划分，同时得到了具有一定性质的向量双弯曲函数、弯曲划分，并在某些条件下得到了拉丁平方偏差分集填充。首先，我们通过向量函数的直接和来分析结构，然后给出广义Maiorana-McFarland结构的一个版本。接下来，我们推广了Wang， Fu, and Wei（2023）的向量双弯曲函数的构造。最后，我们使用Jedwab和Li（2021）提出的lp -packing的提升过程来构造初等阿贝珥群中的向量双弯曲函数、弯曲分区和lp -packing。利用这些构造，可以得到大量的向量弯曲函数、弯曲划分、lp -填料和相关的非对称组合方案。
{"title":"Bent Partition, Vectorial Dual-Bent Function, and LP-Packing Constructions","authors":"Sezel Alkan;Nurdagül Anbar;Tekgül Kalaycı;Wilfried Meidl","doi":"10.1109/TIT.2024.3505600","DOIUrl":"https://doi.org/10.1109/TIT.2024.3505600","url":null,"abstract":"We present secondary constructions of vectorial functions respectively partitions of elementary abelian groups, which simultaneously yield vectorial dual-bent functions with certain properties, bent partitions, and under some conditions, Latin square partial difference set packings (LP-packings). First, we analyse constructions via the direct sum of vectorial functions and then present a version of the generalized Maiorana-McFarland construction. Next, we generalize a construction of vectorial dual-bent functions by Wang, Fu, and Wei (2023). Finally, we use a lifting procedure of LP-packings from Jedwab and Li (2021) to construct vectorial dual-bent functions, bent partitions, and LP-packings in elementary abelian groups. With these constructions, a large variety of vectorial bent functions, bent partitions, LP-packings, and related amorphic association schemes can be obtained.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"752-767"},"PeriodicalIF":2.2,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890198","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 : 2024-11-25DOI: 10.1109/TIT.2024.3493737
{"title":"IEEE Transactions on Information Theory Information for Authors","authors":"","doi":"10.1109/TIT.2024.3493737","DOIUrl":"https://doi.org/10.1109/TIT.2024.3493737","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"C3-C3"},"PeriodicalIF":2.2,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10767126","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142713767","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 : 2024-11-22DOI: 10.1109/TIT.2024.3504538
Vinayak Ramkumar;Myna Vajha;M. Nikhil Krishnan
We study burst erasure correcting streaming codes for three-node relay networks, where there is a source-relay link and a relay-destination link. These codes guarantee that all message packets are recovered within a delay of $tau $ time slots, given that a single burst erasure of length at most b packets occurs in both links. Leveraging previously known techniques in the streaming code literature, we first provide a simple upper bound on the rate of burst erasure correcting streaming codes for three-node relay networks. Our main result is a coding scheme that achieves rates arbitrarily close to the rate upper bound, as message size increases.
{"title":"Streaming Codes for Three-Node Relay Networks With Burst Erasures","authors":"Vinayak Ramkumar;Myna Vajha;M. Nikhil Krishnan","doi":"10.1109/TIT.2024.3504538","DOIUrl":"https://doi.org/10.1109/TIT.2024.3504538","url":null,"abstract":"We study burst erasure correcting streaming codes for three-node relay networks, where there is a source-relay link and a relay-destination link. These codes guarantee that all message packets are recovered within a delay of \u0000<inline-formula> <tex-math>$tau $ </tex-math></inline-formula>\u0000 time slots, given that a single burst erasure of length at most b packets occurs in both links. Leveraging previously known techniques in the streaming code literature, we first provide a simple upper bound on the rate of burst erasure correcting streaming codes for three-node relay networks. Our main result is a coding scheme that achieves rates arbitrarily close to the rate upper bound, as message size increases.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"348-359"},"PeriodicalIF":2.2,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890297","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 : 2024-11-22DOI: 10.1109/TIT.2024.3504823
Yanyan Dong;Shenghao Yang
An $(n,M)$ code refers to a binary code with blocklength n and codebook size M. Such codes are studied in the context of memoryless binary symmetric channels (BSCs) with maximum likelihood (ML) decoding. Previous research has characterized some optimal codes among the linear $(n,4)$ codes for any $n geq 2$ . However, it was unknown whether these optimal codes among linear codes were better than all nonlinear codes. In this paper, we first demonstrate that for any $n geq 2$ , there exists an optimal code among all $(n,4)$ codes that is either linear or belongs to a subset of nonlinear codes called Class-I codes. We identify all the optimal codes among the linear $(n,4)$ codes for each blocklength $n geq 2$ and discover some that were not previously reported in the literature. For any n from 2 to 8, all the optimal $(n,4)$ codes are identified. Except for $n=3$ , all the optimal $(n,4)$ codes are equivalent to linear codes. There exist optimal $(3,4)$ codes that are not equivalent to linear codes. Furthermore, we introduce a subset of nonlinear codes called Class-II codes and show that for any $n gt 3$ , the set composed of linear, Class-I, and Class-II codes and their equivalent codes contains all the optimal $(n,4)$ codes. Both Class-I and Class-II codes are close to linear codes in the sense that they involve only one type of column that is not included in linear codes. We derive a sufficient condition such that all the optimal $(n,4)$ codes are equivalent to linear codes, which can be evaluated by computer with a computation cost $O(n^{6})$ .
{"title":"On Optimal Finite-Length Block Codes of Size Four for Binary Symmetric Channels","authors":"Yanyan Dong;Shenghao Yang","doi":"10.1109/TIT.2024.3504823","DOIUrl":"https://doi.org/10.1109/TIT.2024.3504823","url":null,"abstract":"An \u0000<inline-formula> <tex-math>$(n,M)$ </tex-math></inline-formula>\u0000 code refers to a binary code with blocklength n and codebook size M. Such codes are studied in the context of memoryless binary symmetric channels (BSCs) with maximum likelihood (ML) decoding. Previous research has characterized some optimal codes among the linear \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes for any \u0000<inline-formula> <tex-math>$n geq 2$ </tex-math></inline-formula>\u0000. However, it was unknown whether these optimal codes among linear codes were better than all nonlinear codes. In this paper, we first demonstrate that for any \u0000<inline-formula> <tex-math>$n geq 2$ </tex-math></inline-formula>\u0000, there exists an optimal code among all \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes that is either linear or belongs to a subset of nonlinear codes called Class-I codes. We identify all the optimal codes among the linear \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes for each blocklength \u0000<inline-formula> <tex-math>$n geq 2$ </tex-math></inline-formula>\u0000 and discover some that were not previously reported in the literature. For any n from 2 to 8, all the optimal \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes are identified. Except for \u0000<inline-formula> <tex-math>$n=3$ </tex-math></inline-formula>\u0000, all the optimal \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes are equivalent to linear codes. There exist optimal \u0000<inline-formula> <tex-math>$(3,4)$ </tex-math></inline-formula>\u0000 codes that are not equivalent to linear codes. Furthermore, we introduce a subset of nonlinear codes called Class-II codes and show that for any \u0000<inline-formula> <tex-math>$n gt 3$ </tex-math></inline-formula>\u0000, the set composed of linear, Class-I, and Class-II codes and their equivalent codes contains all the optimal \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes. Both Class-I and Class-II codes are close to linear codes in the sense that they involve only one type of column that is not included in linear codes. We derive a sufficient condition such that all the optimal \u0000<inline-formula> <tex-math>$(n,4)$ </tex-math></inline-formula>\u0000 codes are equivalent to linear codes, which can be evaluated by computer with a computation cost \u0000<inline-formula> <tex-math>$O(n^{6})$ </tex-math></inline-formula>\u0000.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"138-166"},"PeriodicalIF":2.2,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890176","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 : 2024-11-21DOI: 10.1109/TIT.2024.3503717
Kengo Hashimoto;Ken-Ichi Iwata
For an integer $k geq 0$ , k-bit delay decodable code-tuples are source codes that use a finite number of code tables and allow a decoding delay of at most k bits. It is known that the class of k-bit delay decodable code-tuples can achieve a better average codeword length than Huffman codes for $k geq 2$ . However, it is generally challenging to find an optimal k-bit delay decodable code-tuple (i.e., a k-bit delay decodable code-tuple achieving the optimal average codeword length among all k-bit delay decodable code-tuples) because the class of k-bit delay decodable code-tuples is a comprehensive and flexible class containing a variety of source code consisting of any finite number of code tables. AIFV (almost instantaneous fixed-to-variable length) codes are 2-bit delay decodable code-tuples consisting of two code tables satisfying certain constraints. This paper proves that the class of AIFV codes always contains an optimal 2-bit delay decodable code-tuple for any given source distribution. Thus, we can find an optimal 2-bit delay decodable code-tuple in the class of 2-bit delay decodable code-tuples by considering only the class of AIFV codes, which is a very restricted subclass compared to the whole class of 2-bit delay decodable code-tuples.
{"title":"Optimal Codes in the Class of 2-Bit Delay Decodable Codes","authors":"Kengo Hashimoto;Ken-Ichi Iwata","doi":"10.1109/TIT.2024.3503717","DOIUrl":"https://doi.org/10.1109/TIT.2024.3503717","url":null,"abstract":"For an integer \u0000<inline-formula> <tex-math>$k geq 0$ </tex-math></inline-formula>\u0000, k-bit delay decodable code-tuples are source codes that use a finite number of code tables and allow a decoding delay of at most k bits. It is known that the class of k-bit delay decodable code-tuples can achieve a better average codeword length than Huffman codes for \u0000<inline-formula> <tex-math>$k geq 2$ </tex-math></inline-formula>\u0000. However, it is generally challenging to find an optimal k-bit delay decodable code-tuple (i.e., a k-bit delay decodable code-tuple achieving the optimal average codeword length among all k-bit delay decodable code-tuples) because the class of k-bit delay decodable code-tuples is a comprehensive and flexible class containing a variety of source code consisting of any finite number of code tables. AIFV (almost instantaneous fixed-to-variable length) codes are 2-bit delay decodable code-tuples consisting of two code tables satisfying certain constraints. This paper proves that the class of AIFV codes always contains an optimal 2-bit delay decodable code-tuple for any given source distribution. Thus, we can find an optimal 2-bit delay decodable code-tuple in the class of 2-bit delay decodable code-tuples by considering only the class of AIFV codes, which is a very restricted subclass compared to the whole class of 2-bit delay decodable code-tuples.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 1","pages":"797-832"},"PeriodicalIF":2.2,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890232","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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