Pub Date : 2017-05-03DOI: 10.1109/TIT.2017.2700859
A. Barg, Itzhak Tamo, S. Vladut
A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, vol. 60, no. 8, 2014, pp. 4661-4676). In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia-Stichtenoth towers. The local recovery procedure is performed by polynomial interpolation over r coordinates of the codevector. We also obtain a family of Hermitian codes with two disjoint recovering sets for every symbol of the codeword.
如果编码中的每个符号都是少量(最多r个)其他符号的函数，则有限字母表上的代码称为局部可恢复(LRC)代码。在最近的一篇论文中，I. Tamo和A. Barg (IEEE Trans)构造了一类线性LRC码，它们推广了Reed-Solomon码的经典构造。通知。《理论》，第60卷，第6期。8, 2014, pp. 4661-4676)。本文将这种构造推广到代数曲线上的码。我们给出了曲线上LRC码的一般构造，并计算了一些例子，包括从Garcia-Stichtenoth塔得到的渐近好的码族。局部恢复过程是执行多项式插值在r坐标的协矢量。对于码字的每个符号，我们也得到了具有两个不相交恢复集的厄米码族。
{"title":"Locally recoverable codes on algebraic curves","authors":"A. Barg, Itzhak Tamo, S. Vladut","doi":"10.1109/TIT.2017.2700859","DOIUrl":"https://doi.org/10.1109/TIT.2017.2700859","url":null,"abstract":"A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, vol. 60, no. 8, 2014, pp. 4661-4676). In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia-Stichtenoth towers. The local recovery procedure is performed by polynomial interpolation over r coordinates of the codevector. We also obtain a family of Hermitian codes with two disjoint recovering sets for every symbol of the codeword.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128519433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-03-01DOI: 10.1109/TIT.2016.2647724
G. Garrammone, D. Declercq, M. Fossorier
The non-binary codeword weight distribution and its growth rate are developed for non-binary multi-edge type ensembles of low-density parity-check codes. Moreover, an analysis of the growth rate for small weights is provided. The derived expressions can serve as powerful and flexible tools to analyze and design the non-binary low-density parity-check codes that fall within the multi-edge type framework.
{"title":"Weight distributions of non-binary multi-edge type LDPC code ensembles","authors":"G. Garrammone, D. Declercq, M. Fossorier","doi":"10.1109/TIT.2016.2647724","DOIUrl":"https://doi.org/10.1109/TIT.2016.2647724","url":null,"abstract":"The non-binary codeword weight distribution and its growth rate are developed for non-binary multi-edge type ensembles of low-density parity-check codes. Moreover, an analysis of the growth rate for small weights is provided. The derived expressions can serve as powerful and flexible tools to analyze and design the non-binary low-density parity-check codes that fall within the multi-edge type framework.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122907430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-14DOI: 10.1109/TIT.2016.2580545
Varun Jog, V. Anantharam
We consider the additive white Gaussian noise (AWGN) channel with a (σ, ρ)-power constraint, which is motivated by energy harvesting communication systems. This constraint imposes a limit of σ + kρ on the total power of any k ≥ 1 consecutive transmitted symbols in a codeword. We analyze the capacity of this channel geometrically, by considering the set Sn(σ, ρ) ⊆ ℝn which is the set of all n-length sequences satisfying the (σ, ρ)-power constraints. For a noise power of ν, we obtain an upper bound on capacity by considering the volume of the Minkowski sum of Sn(σ, ρ) and the n-dimensional Euclidean ball of radius √(nν). We analyze this bound using a result from convex geometry known as Steiner's formula, which gives the volume of this Minkowski sum in terms of the intrinsic volumes of Sn(σ, ρ). We show that as n increases, the logarithms of the intrinsic volumes of {Sn(σ, ρ)} converge to a limit function under an appropriate scaling. An upper bound on capacity is obtained in terms of the limit function, thus pinning down the asymptotic capacity of the (σ, ρ)-power constrained AWGN channel in the low-noise regime. We derive stronger results when σ = 0, corresponding to the amplitude-constrained AWGN channel.
{"title":"A geometric analysis of the AWGN channel with a (σ, ρ)-power constraint","authors":"Varun Jog, V. Anantharam","doi":"10.1109/TIT.2016.2580545","DOIUrl":"https://doi.org/10.1109/TIT.2016.2580545","url":null,"abstract":"We consider the additive white Gaussian noise (AWGN) channel with a (σ, ρ)-power constraint, which is motivated by energy harvesting communication systems. This constraint imposes a limit of σ + kρ on the total power of any k ≥ 1 consecutive transmitted symbols in a codeword. We analyze the capacity of this channel geometrically, by considering the set S<sub>n</sub>(σ, ρ) ⊆ ℝ<sup>n</sup> which is the set of all n-length sequences satisfying the (σ, ρ)-power constraints. For a noise power of ν, we obtain an upper bound on capacity by considering the volume of the Minkowski sum of S<sub>n</sub>(σ, ρ) and the n-dimensional Euclidean ball of radius √(nν). We analyze this bound using a result from convex geometry known as Steiner's formula, which gives the volume of this Minkowski sum in terms of the intrinsic volumes of S<sub>n</sub>(σ, ρ). We show that as n increases, the logarithms of the intrinsic volumes of {S<sub>n</sub>(σ, ρ)} converge to a limit function under an appropriate scaling. An upper bound on capacity is obtained in terms of the limit function, thus pinning down the asymptotic capacity of the (σ, ρ)-power constrained AWGN channel in the low-noise regime. We derive stronger results when σ = 0, corresponding to the amplitude-constrained AWGN channel.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114591381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282457
Andrew J. Young, Yury Polyanskiy
This article continues the recent investigation of combinatorial joint source-channel coding. For the special case of a binary source and channel subject to distortion measured by Hamming distance, the lower (converse) bounds on achievable source distortion are improved for all values of channel noise. Operational duality between coding with bandwidth expansion factors ρ and 1 over ρ is established. Although the exact value of the asymptotic noise-distortion tradeoff curve is unknown (except at ρ = 1), some initial results on inter-relations between these curves for different values of ρ are shown and lead to statements about monotonicity and continuity in ρ.
{"title":"Converse and duality results for combinatorial source-channel coding in binary Hamming spaces","authors":"Andrew J. Young, Yury Polyanskiy","doi":"10.1109/ISIT.2015.7282457","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282457","url":null,"abstract":"This article continues the recent investigation of combinatorial joint source-channel coding. For the special case of a binary source and channel subject to distortion measured by Hamming distance, the lower (converse) bounds on achievable source distortion are improved for all values of channel noise. Operational duality between coding with bandwidth expansion factors ρ and 1 over ρ is established. Although the exact value of the asymptotic noise-distortion tradeoff curve is unknown (except at ρ = 1), some initial results on inter-relations between these curves for different values of ρ are shown and lead to statements about monotonicity and continuity in ρ.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"192 S539","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113972692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282519
Shwan Ashrafi, Chen Feng, Sumit Roy, F. Kschischang
The benefit of applying compute-and-forward (C&F) to slotted ALOHA (S-ALOHA) systems is studied. A Markov chain model is introduced, and an approximate stability region is given. It is shown that the approximate region is asymptotically exact as the number of users tends to infinity. It is also shown that the approximate region is very accurate even for systems with a small number of users. Further, based on the approximate region, simple expressions for the throughput and delay performance of S-ALOHA with C&F are derived, demonstrating the significant advantages offered by C&F.
{"title":"Slotted ALOHA with compute-and-forward","authors":"Shwan Ashrafi, Chen Feng, Sumit Roy, F. Kschischang","doi":"10.1109/ISIT.2015.7282519","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282519","url":null,"abstract":"The benefit of applying compute-and-forward (C&F) to slotted ALOHA (S-ALOHA) systems is studied. A Markov chain model is introduced, and an approximate stability region is given. It is shown that the approximate region is asymptotically exact as the number of users tends to infinity. It is also shown that the approximate region is very accurate even for systems with a small number of users. Further, based on the approximate region, simple expressions for the throughput and delay performance of S-ALOHA with C&F are derived, demonstrating the significant advantages offered by C&F.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122539109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282705
M. Khojastepour, Mohammad Farajzadeh-Tehrani
We consider the classical problem of spatial interference alignment (IA) in MIMO channels with constant channel coefficients through design of linear transmit precoders and receiver filters. Some easily (polynomial time) computable necessary conditions for IA have been derived in the literature [1], [2], [3]. Computable sufficient and necessary conditions that completely characterizes the feasibility of an IA problem have also been obtained [4], [2]. However, it has been shown that checking the feasibility of interference alignment when the number of antennas are more than two is NP-complete[3]. This result is inline with full characterization of the feasibility of IA as the sufficiency conditions require multiplication of Schubert cycles that becomes exhaustive as the dimensions grows. Naturally, the following questions may arise: “Is it possible to have a sufficiency condition for a general case of IA based on only the dimensions of the system (number of antennas at each node and degrees of freedom (DoF) per node [4]) that is simple (polynomial time) to compute?” and “How effective such sufficiency conditions would be?”. In this paper, we provide an affirmative answer to the first question and show the proposed sufficient condition is asymptotically optimal. The sufficiency conditions are expressed in terms of simple inequalities based on system dimensions. Unlike necessary conditions that are based on simple argument such as dimension counting [1], we have not yet been able to provide an elementary proof for the derived sufficiency conditions. The provided proof requires familiarity with Schubert calculus over complex Grassmannians.
{"title":"A sufficient condition for interference alignment","authors":"M. Khojastepour, Mohammad Farajzadeh-Tehrani","doi":"10.1109/ISIT.2015.7282705","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282705","url":null,"abstract":"We consider the classical problem of spatial interference alignment (IA) in MIMO channels with constant channel coefficients through design of linear transmit precoders and receiver filters. Some easily (polynomial time) computable necessary conditions for IA have been derived in the literature [1], [2], [3]. Computable sufficient and necessary conditions that completely characterizes the feasibility of an IA problem have also been obtained [4], [2]. However, it has been shown that checking the feasibility of interference alignment when the number of antennas are more than two is NP-complete[3]. This result is inline with full characterization of the feasibility of IA as the sufficiency conditions require multiplication of Schubert cycles that becomes exhaustive as the dimensions grows. Naturally, the following questions may arise: “Is it possible to have a sufficiency condition for a general case of IA based on only the dimensions of the system (number of antennas at each node and degrees of freedom (DoF) per node [4]) that is simple (polynomial time) to compute?” and “How effective such sufficiency conditions would be?”. In this paper, we provide an affirmative answer to the first question and show the proposed sufficient condition is asymptotically optimal. The sufficiency conditions are expressed in terms of simple inequalities based on system dimensions. Unlike necessary conditions that are based on simple argument such as dimension counting [1], we have not yet been able to provide an elementary proof for the derived sufficiency conditions. The provided proof requires familiarity with Schubert calculus over complex Grassmannians.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122647621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282684
F. Etezadi, A. Khisti, Jun Chen
We study sequential transmission of Gauss-Markov sources over erasure channels under a zero decoding delay constraint. A two-stage coding scheme which can be described as a hybrid between predictive coding with limited past and quantization & binning is proposed. This scheme can achieve significant performance gains over baseline schemes in simulations involving i.i.d. erasure channels, and in certain regimes can attain performance close to a fundamental lower bound. We consider an information theoretic model for streaming that explains the weakness of baseline schemes (e.g., predictive coding, memoryless binning, etc.) and illustrates the advantage of our proposed hybrid scheme over these. Techniques from multi-terminal source coding are used to derive a new lower bound on the compression rate and identify cases when the hybrid coding scheme is close to optimal. We discuss qualitatively the interplay between the parameters of our information theoretic model and the statistical models used in simulations.
{"title":"Price of perfection: Limited prediction for streaming over erasure channels","authors":"F. Etezadi, A. Khisti, Jun Chen","doi":"10.1109/ISIT.2015.7282684","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282684","url":null,"abstract":"We study sequential transmission of Gauss-Markov sources over erasure channels under a zero decoding delay constraint. A two-stage coding scheme which can be described as a hybrid between predictive coding with limited past and quantization & binning is proposed. This scheme can achieve significant performance gains over baseline schemes in simulations involving i.i.d. erasure channels, and in certain regimes can attain performance close to a fundamental lower bound. We consider an information theoretic model for streaming that explains the weakness of baseline schemes (e.g., predictive coding, memoryless binning, etc.) and illustrates the advantage of our proposed hybrid scheme over these. Techniques from multi-terminal source coding are used to derive a new lower bound on the compression rate and identify cases when the hybrid coding scheme is close to optimal. We discuss qualitatively the interplay between the parameters of our information theoretic model and the statistical models used in simulations.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115226205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282654
Erdem Koyuncu, H. Jafarkhani
We introduce and investigate the opportunities of multi-antenna communication schemes whose training and feedback stages are interleaved and mutually interacting. Specifically, unlike the traditional schemes where the transmitter first trains all of its antennas at once and then receives a single feedback message, we consider a scenario where the transmitter instead trains its antennas one by one and receives feedback information immediately after training each one of its antennas. The feedback message may ask the transmitter to train another antenna; or, it may terminate the feedback/training phase and provide the quantized codeword (e.g., a beamforming vector) to be utilized for data transmission. As a specific application, we consider a multiple-input single-output system with t transmitter antennas, a short-term power constraint P, and target data rate ρ. We show that for any t, the same outage probability as a system with perfect transmitter and receiver channel state information can be achieved with a feedback rate of R1 bits per channel state and via training R2 transmitter antennas on average, where R1 and R2 are independent of t, and depend only on ρ and P.
{"title":"Interleaving training and limited feedback for point-to-point massive multiple-antenna systems","authors":"Erdem Koyuncu, H. Jafarkhani","doi":"10.1109/ISIT.2015.7282654","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282654","url":null,"abstract":"We introduce and investigate the opportunities of multi-antenna communication schemes whose training and feedback stages are interleaved and mutually interacting. Specifically, unlike the traditional schemes where the transmitter first trains all of its antennas at once and then receives a single feedback message, we consider a scenario where the transmitter instead trains its antennas one by one and receives feedback information immediately after training each one of its antennas. The feedback message may ask the transmitter to train another antenna; or, it may terminate the feedback/training phase and provide the quantized codeword (e.g., a beamforming vector) to be utilized for data transmission. As a specific application, we consider a multiple-input single-output system with t transmitter antennas, a short-term power constraint P, and target data rate ρ. We show that for any t, the same outage probability as a system with perfect transmitter and receiver channel state information can be achieved with a feedback rate of R1 bits per channel state and via training R2 transmitter antennas on average, where R1 and R2 are independent of t, and depend only on ρ and P.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115417651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-10-01DOI: 10.1109/ISIT.2015.7282507
V. Chandar, A. Tchamkerten
In a recently proposed asynchronous communication setup, the receiver observes mostly pure background noise except for a brief and a priori unknown period of time when data is transmitted. Capacity per unit cost and minimum communication delay were characterized and shown to be unaffected by a sparse sampling at the receiver as long as the number of samples represents a constant fraction of the total channel outputs.
{"title":"Asynchronous capacity per unit cost under a receiver sampling constraint","authors":"V. Chandar, A. Tchamkerten","doi":"10.1109/ISIT.2015.7282507","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282507","url":null,"abstract":"In a recently proposed asynchronous communication setup, the receiver observes mostly pure background noise except for a brief and a priori unknown period of time when data is transmitted. Capacity per unit cost and minimum communication delay were characterized and shown to be unaffected by a sparse sampling at the receiver as long as the number of samples represents a constant fraction of the total channel outputs.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116142253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-06-14DOI: 10.1109/ISIT.2015.7282449
Hsien-Ping Lin, Shu Lin, K. Abdel-Ghaffar
Polar codes proposed by Arikan are based on a linear kernel of dimension two with exponent 0.5. In this paper, binary kernels of maximum exponents of dimensions up to 16 are presented except for the case of dimension 12 where the maximum exponent is shown to be attained by either a constructed linear kernel or a possible nonlinear kernel with a specified partial distance sequence. The results show that the minimum dimension for which there exists a kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arikan, is 14. For dimensions 14, 15, 16, discussed by Presman et al., along with 13, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.
{"title":"Binary nonlinear kernels of maximum exponents of polar codes of dimensions up to sixteen","authors":"Hsien-Ping Lin, Shu Lin, K. Abdel-Ghaffar","doi":"10.1109/ISIT.2015.7282449","DOIUrl":"https://doi.org/10.1109/ISIT.2015.7282449","url":null,"abstract":"Polar codes proposed by Arikan are based on a linear kernel of dimension two with exponent 0.5. In this paper, binary kernels of maximum exponents of dimensions up to 16 are presented except for the case of dimension 12 where the maximum exponent is shown to be attained by either a constructed linear kernel or a possible nonlinear kernel with a specified partial distance sequence. The results show that the minimum dimension for which there exists a kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arikan, is 14. For dimensions 14, 15, 16, discussed by Presman et al., along with 13, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.","PeriodicalId":272313,"journal":{"name":"2015 IEEE International Symposium on Information Theory (ISIT)","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115146224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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