Pub Date : 2013-02-01DOI: 10.1109/TIT.2012.2219154
Urs Niesen, S. Diggavi
We consider the Gaussian “diamond” or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log4(N)) of capacity with preconstant in the order notation independent of the channel gains.
{"title":"The Approximate Capacity of the Gaussian $N$-Relay Diamond Network","authors":"Urs Niesen, S. Diggavi","doi":"10.1109/TIT.2012.2219154","DOIUrl":"https://doi.org/10.1109/TIT.2012.2219154","url":null,"abstract":"We consider the Gaussian “diamond” or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log4(N)) of capacity with preconstant in the order notation independent of the channel gains.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"17 1","pages":"845-859"},"PeriodicalIF":0.0,"publicationDate":"2013-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83907869","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 : 2009-05-01DOI: 10.1109/TIT.2009.2016030
Mike Davies, Rémi Gribonval
This paper investigates conditions under which the solution of an underdetermined linear system with minimal lP norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Matrices are constructed with restricted isometry constants (RIC) δ2m arbitrarily close to 1/ √2 ≅ 0.707 where sparse recovery with p = 1 fails for at least one m-sparse vector, as well as matrices with δ2m arbitrarily close to one where l1 minimization succeeds for any m-sparse vector. This highlights the pessimism of sparse recovery prediction based on the RIC, and indicates that there is limited room for improving over the best known positive results of Foucart and Lai, which guarantee that l1 minimization recovers all m-sparse vedors for any matrix with δ2m < 2(3 - √2)/7 ≅ 0.4531. These constructions are a by-product of tight conditions for lp recovery (0 ≤ p ≤ 1) with matrices of unit spectral norm, which are expressed in terms of the minimal singular values of 2m-column submatrices. Compared to l1 minimization, lp minimization recovery failure is shown to be only slightly delayed in terms of the RIC values. Furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used. It is shown that when lp optimization is attempted using an iterative reweighted l1 scheme, failure can still occur for δ2m arbitrarily close to 1/ √2.
{"title":"Restricted isometry constants where lpsparse recovery can fail for 0 < p <= 1","authors":"Mike Davies, Rémi Gribonval","doi":"10.1109/TIT.2009.2016030","DOIUrl":"https://doi.org/10.1109/TIT.2009.2016030","url":null,"abstract":"This paper investigates conditions under which the solution of an underdetermined linear system with minimal lP norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Matrices are constructed with restricted isometry constants (RIC) δ2m arbitrarily close to 1/ √2 ≅ 0.707 where sparse recovery with p = 1 fails for at least one m-sparse vector, as well as matrices with δ2m arbitrarily close to one where l1 minimization succeeds for any m-sparse vector. This highlights the pessimism of sparse recovery prediction based on the RIC, and indicates that there is limited room for improving over the best known positive results of Foucart and Lai, which guarantee that l1 minimization recovers all m-sparse vedors for any matrix with δ2m < 2(3 - √2)/7 ≅ 0.4531. These constructions are a by-product of tight conditions for lp recovery (0 ≤ p ≤ 1) with matrices of unit spectral norm, which are expressed in terms of the minimal singular values of 2m-column submatrices. Compared to l1 minimization, lp minimization recovery failure is shown to be only slightly delayed in terms of the RIC values. Furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used. It is shown that when lp optimization is attempted using an iterative reweighted l1 scheme, failure can still occur for δ2m arbitrarily close to 1/ √2.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"5 1","pages":"2203-2214"},"PeriodicalIF":0.0,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88624445","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}
{"title":"Correction to \"On the Capacity Game of Private Fingerprinting Systems Under Collusion Attacks\" [Mar 05 884-899]","authors":"A. Somekh-Baruch, N. Merhav","doi":"10.1109/TIT.2008.929955","DOIUrl":"https://doi.org/10.1109/TIT.2008.929955","url":null,"abstract":"In this correspondence, we correct an error in the above paper by Somekh-Baruch and Merhav (see ibid., vol.51, no.3, p.884-99, 2005).","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"4 1","pages":"5263-5264"},"PeriodicalIF":0.0,"publicationDate":"2008-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82661532","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}
For original paper see S. Ray, ibid., vol 53, no. 6, pp. 1983-2009, 2007. The multiple-input multiple-output (MIMO) wideband Rayleigh block-fading channel where the channel state is unknown to both the transmitter and the receiver and there is only an average power constraint on the input. This mutual information can be upper bounded by the capacity of AWGN channel with the same power constraint.
原论文见S. Ray，同上，第53卷，no. 5。6, pp. 1983-2009, 2007。多输入多输出(MIMO)宽带瑞利块衰落信道，该信道的状态对发送方和接收方都是未知的，并且在输入端只存在平均功率约束。这种互信息可以以具有相同功率约束的AWGN信道容量为上界。
{"title":"Correction to \"On Noncoherent MIMO Channels in the Wideband Regime: Capacity and Reliability\" [Jun 07 1983-2009]","authors":"Surapol Tan-a-ram","doi":"10.1109/TIT.2008.929921","DOIUrl":"https://doi.org/10.1109/TIT.2008.929921","url":null,"abstract":"For original paper see S. Ray, ibid., vol 53, no. 6, pp. 1983-2009, 2007. The multiple-input multiple-output (MIMO) wideband Rayleigh block-fading channel where the channel state is unknown to both the transmitter and the receiver and there is only an average power constraint on the input. This mutual information can be upper bounded by the capacity of AWGN channel with the same power constraint.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"289 1","pages":"5263"},"PeriodicalIF":0.0,"publicationDate":"2008-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76875642","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}
For original article by C.V. Chong see ibid. Vol.49, no.11, p.2953-2959, Nov. 2003. Some corrections are given for the sequence pairing descriptions of 16-QAM Golay complementary sequences in Chong, Venkataramani, and Tarokh's paper, together with a related correction for Lee and Golomb's 64-QAM Golay sequence construction. Lee and Golomb obtained 496, 808, and 976 offset pairs for length 2m 64-QAM Golay sequences, m = 2,3,4. We obtained 724, 972, and 1224 offset pairs. Adding w = 1 to Case III in Lee and Golomb's construction gives some additional offset pairs, others are new and exist for m ges 3 only. Descriptions of new offset pairs and enumeration for all first order offset pairs are proposed as conjectures. An example is given to show that there exist other 64-QAM Golay sequences not within this construction.
{"title":"Comments on \"A New Construction of 16-QAM Golay Complementary Sequences\" and Extension for 64-QAM Golay Sequences","authors":"Ying Li","doi":"10.1109/TIT.2008.924735","DOIUrl":"https://doi.org/10.1109/TIT.2008.924735","url":null,"abstract":"For original article by C.V. Chong see ibid. Vol.49, no.11, p.2953-2959, Nov. 2003. Some corrections are given for the sequence pairing descriptions of 16-QAM Golay complementary sequences in Chong, Venkataramani, and Tarokh's paper, together with a related correction for Lee and Golomb's 64-QAM Golay sequence construction. Lee and Golomb obtained 496, 808, and 976 offset pairs for length 2m 64-QAM Golay sequences, m = 2,3,4. We obtained 724, 972, and 1224 offset pairs. Adding w = 1 to Case III in Lee and Golomb's construction gives some additional offset pairs, others are new and exist for m ges 3 only. Descriptions of new offset pairs and enumeration for all first order offset pairs are proposed as conjectures. An example is given to show that there exist other 64-QAM Golay sequences not within this construction.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"30 1","pages":"3246-3251"},"PeriodicalIF":0.0,"publicationDate":"2008-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82380696","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}
A new generalization of the Gray map is introduced. The new generalization Phi:Z2 kn rarr Z2 2k-1n is connected with the known generalized Gray map phi in the following way: if we take two dual linear Z2 k-codes and construct binary codes from them using the generalizations phi and Phi of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of Z2 k-linear Hadamard codes and co-Z2 k-linear extended 1-perfect codes are described, where co-Z2 k-linearity means that the code can be obtained from a linear Z2 k-code with the help of the new generalized Gray map
{"title":"On Z2k-Dual Binary Codes","authors":"D. Krotov","doi":"10.1109/TIT.2007.892787","DOIUrl":"https://doi.org/10.1109/TIT.2007.892787","url":null,"abstract":"A new generalization of the Gray map is introduced. The new generalization Phi:Z2 kn rarr Z2 2k-1n is connected with the known generalized Gray map phi in the following way: if we take two dual linear Z2 k-codes and construct binary codes from them using the generalizations phi and Phi of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of Z2 k-linear Hadamard codes and co-Z2 k-linear extended 1-perfect codes are described, where co-Z2 k-linearity means that the code can be obtained from a linear Z2 k-code with the help of the new generalized Gray map","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"46 3 1","pages":"1532-1537"},"PeriodicalIF":0.0,"publicationDate":"2005-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79909572","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}
We present a new construction of 16-QAM Golay sequences of length n = 2/sup m/. The number of constructed sequences is (14 + 12m)(m!/2)4/sup m+1/. When employed as a code in an orthogonal frequency-division multiplexing (OFDM) system; this set of sequences has a peak-to-mean envelope power ratio (PMEPR) of 3.6. By considering two specific subsets of these sequences, we obtain new codes with PMEPR bounds of 2.0 and 2.8 and respective code sizes of (2 + 2m)(m!/2)4/sup m+1/ and (4 + 4m)(m!/2)4/sup m+1/. These are larger than previously known codes for the same PMEPR bounds.
{"title":"A new construction of 16-QAM Golay complementary sequences","authors":"C. Chong, R. Venkataramani, V. Tarokh","doi":"10.1109/TIT.2003.818418","DOIUrl":"https://doi.org/10.1109/TIT.2003.818418","url":null,"abstract":"We present a new construction of 16-QAM Golay sequences of length n = 2/sup m/. The number of constructed sequences is (14 + 12m)(m!/2)4/sup m+1/. When employed as a code in an orthogonal frequency-division multiplexing (OFDM) system; this set of sequences has a peak-to-mean envelope power ratio (PMEPR) of 3.6. By considering two specific subsets of these sequences, we obtain new codes with PMEPR bounds of 2.0 and 2.8 and respective code sizes of (2 + 2m)(m!/2)4/sup m+1/ and (4 + 4m)(m!/2)4/sup m+1/. These are larger than previously known codes for the same PMEPR bounds.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"79 1","pages":"2953-2959"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81417224","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}
A code structure is introduced that represents a Reed-Solomon (RS) code in two-dimensional format. Based on this structure, a novel approach to multiple error burst correction using RS codes is proposed. For a model of phased error bursts, where each burst can affect one of the columns in a two-dimensional transmitted word, it is shown that the bursts can be corrected using a known multisequence shift-register synthesis algorithm. It is further shown that the resulting codes posses nearly optimal burst correction capability, under certain probability of decoding failure. Finally, low-complexity systematic encoding and syndrome computation algorithms for these codes are discussed. The proposed scheme may also find use in decoding of different coding schemes based on RS codes, such as product or concatenated codes.
{"title":"Reed-Solomon codes for correcting phased error bursts","authors":"V. Krachkovsky","doi":"10.1109/TIT.2003.819333","DOIUrl":"https://doi.org/10.1109/TIT.2003.819333","url":null,"abstract":"A code structure is introduced that represents a Reed-Solomon (RS) code in two-dimensional format. Based on this structure, a novel approach to multiple error burst correction using RS codes is proposed. For a model of phased error bursts, where each burst can affect one of the columns in a two-dimensional transmitted word, it is shown that the bursts can be corrected using a known multisequence shift-register synthesis algorithm. It is further shown that the resulting codes posses nearly optimal burst correction capability, under certain probability of decoding failure. Finally, low-complexity systematic encoding and syndrome computation algorithms for these codes are discussed. The proposed scheme may also find use in decoding of different coding schemes based on RS codes, such as product or concatenated codes.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"283 1","pages":"2975-2984"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79538205","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}
A constant-composition code is a special constant-weight code under the restriction that each symbol should appear a given number of times in each codeword. In this correspondence, we give a lower bound for the maximum size of the q-ary constant-composition codes with minimum distance at least 3. This bound is asymptotically optimal and generalizes the Graham-Sloane bound for binary constant-weight codes. In addition, three construction methods of constant-composition codes are presented, and a number of optimum constant-composition codes are obtained by using these constructions.
{"title":"On constant-composition codes over Zq","authors":"Luo Yuan, Fang-Wei Fu, A. Vinck, Wende Chen","doi":"10.1109/TIT.2003.819339","DOIUrl":"https://doi.org/10.1109/TIT.2003.819339","url":null,"abstract":"A constant-composition code is a special constant-weight code under the restriction that each symbol should appear a given number of times in each codeword. In this correspondence, we give a lower bound for the maximum size of the q-ary constant-composition codes with minimum distance at least 3. This bound is asymptotically optimal and generalizes the Graham-Sloane bound for binary constant-weight codes. In addition, three construction methods of constant-composition codes are presented, and a number of optimum constant-composition codes are obtained by using these constructions.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"111 1","pages":"3010-3016"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79622744","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}
We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.
{"title":"On Z4-linear Preparata-like and Kerdock-like code","authors":"J. Borges, K. Phelps, J. Rifà, V. Zinoviev","doi":"10.1109/TIT.2003.819329","DOIUrl":"https://doi.org/10.1109/TIT.2003.819329","url":null,"abstract":"We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"3 1","pages":"2834-2843"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73952129","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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