Classification of poset-block spaces admitting MacWilliams-type identity

In this work we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity we explicit the relation between the weight enumerators of a code and its dual.Comment: 8 pages, 1 figure. Submitted to IEEE Transactions on Information Theor

Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures

Coding in the Presence of Semantic Value of Information: Unequal Error Protection Using Poset Decoders

In this work we explore possibilities for coding when information worlds have different (semantic) values. We introduce a loss function that expresses the overall performance of a coding scheme for discrete channels and exchange the usual goal of minimizing the error probability to that of minimizing the expected loss. In this environment we explore the possibilities of using poset-decoders to make a message-wise unequal error protection (UEP), where the most valuable information is protected by placing in its proximity information words that differ by small valued information. Similar definitions and results are shortly presented also for signal constellations in Euclidean space

Teoria de códigos : uma abordagem usando métricas que respeitam suporte e outros problemas

Orientador: Marcelo FirerTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Esta tese estuda as métricas definidas por peso e respeitam o suporte dos vetores (TS-métricas) no contexto de teoria de códigos. Sua principal preocupação, considerando famílias específicas de métricas, é explorar e entender alguns resultados "estruturais" das métricas, a saber: descrever o grupo de isometrias lineares e, estabelecer condições para a validade da Identidade MacWilliams (uma relação entre a distribuição de peso de um código e a distribuição de peso - possivelmente de um peso modificado - do código dual) e da propriedade de extensão MacWilliams (quando isometrias lineares entre códigos lineares podem ser estendidas para isometrias lineares em todo o espaço). Esses resultados são os primeiros explorados para a família das métricas combinatoriais de Gabidulin e para uma nova família dessas métricas, as métricas de posets-bloco-rotulados. Além disso, é apresentada uma abordagem sistemática ao espaço de todas as TS-métricas, primeiro rotulando as arestas do cubo de Hamming. Em seguida, introduzimos um operador condicional nas TS-métricas, o que permite mostrar que qualquer TS-métrica pode ser obtida por uma sequência de somas condicionais de poset ou métricas combinatórias. Além deste estudo sistemático de TS-métricas, apresentamos alguns resultados relevantes em relação à representação de dígrafosAbstract: This thesis studies metrics determined by weight that are compatible with support of vectors (TS-metrics) in the context of coding theory. Its main concern, considering specific families of metrics, is to explore and understand some ''structural'' results of the metrics, namely: to describe the group of linear isometries and to establish conditions for the validity of the MacWilliams Identity (a relation between the weight distribution of a code and the weight distribution - possibly of a modified weight - of the dual code) and the MacWilliams extension property (when linear isometries between linear codes can be extended to linear isometries on the whole space).These results are first explored for the family of Gabidulin¿s comibinatorial metrics and for a new family of such metrics, the labeled-poset-block metrics. In addition, it is introduce a systematic approach to the space of all TS-metrics, first by labeling the edges of the Hamming cube. Then, we introduce a conditional operator on TS-metrics, which allows to show that any TS-metrics can be obtained, by a sequence of conditional sums of poset or combinatorial metrics. Besides this systematic study of TS-metrics, we present some relevant results concerning representation of digraphsDoutoradoMatematicaDoutor em Matemática2015/11286-8FAPES

Coding and decoding schemes tailor made for image transmission

In this work we explore possibilities for coding and decoding tailor made for image transmission. To do so, we introduce a loss function that expresses the overall performace of a coding scheme for discrete channels and exchange the usual goal of minimizing the error probability to that of minimizing the expected loss. In this environment we explore the possibilities of using poset-decoders to make a message-wise unequal error protection (UEP), where the most valuable information is protected by placing in its proximity information words that differ by small valued information. We give explicit examples, done for scale-of-gray images, including visual simulations for the BSMC. © 2013 IEEE.In this work we explore possibilities for coding and decoding tailor made for image transmission. To do so, we introduce a loss function that expresses the overall performace of a coding scheme for discrete channels and exchange the usual goal of minimiziFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2007/ 56052-8Barg, A., Purkayastha, P., Bounds on ordered codes and orthogonal arrays (2009) Moscow Mathematical Journal, 9 (2), pp. 211-243Berlekamp, E.R., McEliece, R.J., Van Tilborg, H.C.A., On the Inherent Intractability of Certain Coding Problems (1978) IEEE Transactions on Information Theory, 24 (3)Borade, S., Nakiboǧlu, B., Zheng, L., Unequal error protection: An information-theoretic perspective (2009) IEEE Transactions on Information Theory, 55 (12), pp. 5511-5539Brualdi, R., Graves, J.S., Lawrence, M., Codes with a poset metric (1995) Discrete Mathematics, 147, pp. 57-72Cover, T., Thomas, J.A., (2006) Elements of Information Theory, , John Wiley & SonsFirer, M., Panek, L., Ramos Rifo, L.L., (2011) Coding in the Presence of Semantic Value of Information: Unequal Error Protection Using Poset Decoders, , arXiv:1108.3832 [cs.IT]Felix, L.V., Firer, M., Canonical-Systematic Form of Hierarchical Codes (2012) Advances in Mathematics of Communications, 6 (3), pp. 315-328Po Ho, K., Kahn, J.M., Image Transmission over Noisy Channels Using Multicarrier Modulation (1997) Signal Processing: Image Communication, 9, pp. 159-169Hyun, J.Y., Kim, H.K., The poset structures admitting the extended binary Hamming code to be a perfect code (2004) Discrete Mathematics, 288, pp. 37-47Hyun, J.Y., Kim, H.K., Maximum distance separable poset codes (2008) Designs, Codes and Cryptography, 48 (3), pp. 247-261Jang, C., Kim, H.K., Oh, D.Y., Rho, Y., The poset structures admitting the extended binary Golay code to be a perfect code (2008) Discrete Mathematics, 308, pp. 4057-4068Kim, H.K., Oh, D.Y., A Classification of Poset Admitting the MacWilliams Identity (2005) IEEE Transactions on Information Theory, 51 (4), pp. 1424-1431Kolmogorov, A.N., On the Shannon theory of information in the case of continuous signals (1956) IEEE Transactions on Information Theory, , SeptemberMasnick, B., Wolf, J., On linear unequal error protection codes (1967) IEEE Transactions on Information Theory, 3 (4), pp. 600-607De Oliveira Moura, A., Firer, M., Duality for poset codes (2010) IEEE Transactions on Information Theory, 56 (7), pp. 3180-3186Panek, L., Firer, M., Kim, H.K., Hyun, J.Y., Groups of linear isometries on poset structures (2008) Discrete Mathematics, 308, pp. 4116-4123Panek, L., Firer, M., Alves, M.M.S., Symmetry groups of Rosenbloom-Tsfasman spaces (2009) Discrete Mathematics, 309, pp. 763-771Panek, L., Firer, M., Alves, M.M.S., Classification of Niederreiter-Rosenbloom-Tsfasman Block Codes (2010) IEEE Transactions on Information Theory, 56 (10), pp. 5207-5216Pinheiro, J.A., Firer, M., Classification of poset-block spaces admitting MacWilliams-type identity Information Theory, IEEE Transactions on, PP (99), p. 1Parmigiani, G., Inoue, L.Y., (2009) Decision Theory, Principles and Approaches, , Wiley, Cornwall, UKRosenbloom, M.Y., Tsfasman, M.A., Codes for the m-metric (1997) Problems of Information Transmission, 33, pp. 45-52Shannon, C.E., A mathematical theory of communication (1948) Bell System Tech. J., 27, pp. 379-423Shannon, C.E., Coding theorems for a discrete source with a fidelity criterion (1959) IRE Nat. Conv. Rec., , marchSchröder, B.S.W., (2003) Ordered Sets - An Introduction, , BirkhäuserTavildar, S., Viswanath, P., Approximately universal codes over slowfading channels (2006) IEEE Transactions on Information Theory, 52 (7), pp. 3233-3258Information Theory and Applications Workshop (ITA

Bayes decoding for discrete linear channels with semantic value of information

In this work we explore the decoding process when information words have different semantic values. In that setting, we introduce a convenient loss function that allows us to express the overall performance of a decoding scheme for discrete channels, switching the usual goal of minimizing the error probability to that of minimizing the expected loss. © 2012 American Institute of Physics.In this work we explore the decoding process when information words have different semantic values. In that setting, we introduce a convenient loss function that allows us to express the overall performance of a decoding scheme for discrete channels, swit14901126134sem informaçãosem informaçãoAnderson, J., Firer, M., (2011) Classification of poset-block spaces admitting MacWilliams-type identity -preprintAsh, R., Information Theory (1965) John Wiley & Sons, Inc.Barg, A., Purkayastha, P., Bounds on ordered codes and orthogonal arrays (2009) MoscowMathematical Journal, 9 (2), pp. 211-243Berlekamp, E.R., Mceliece, R.J., Tilborg, H.C.A.V., On the Inherent Intractability of Certain Coding Problems (1978) IEEE Transactions on Information Theory, 24 (3)Borade, S., Nakiboǧlu, B., Zheng, L., Unequal error protection: an information-theoretic perspective (2009) IEEE Transactions on Information Theory, 55 (12), pp. 5511-5539Brualdi, R., Graves, J.S., Lawrence, M., Codes with a poset metric (1995) Discrete Mathematics, 147, pp. 57-72Cover, T., Thomas, J.A., Elements of Information Theory (2006) John Wiley & SonsFelix, L.V., Firer, M., Canonical-Systematic Form of Hierarchical Codes (2011) pre-printHo, K.P., Kahn, J.M., Imagen Transmission over Noisy Channels Using Multicarrier Modulation (1997) Signal Processing: Image Comunication, 9, pp. 159-169Hyun, J.Y., Kim, H.K., The poset structures admitting the extended binary Hamming code to be a perfect code (2004) Discrete Mathematics, 288, pp. 37-47Hyun, J.Y., Kim, H.K., Maximum distance separable poset codes (2008) Designs, Codes and Cryptography, 48 (3), pp. 247-261Jang, C., Kim, H.K., Oh, D.Y., Rho, Y., The poset structures admitting the extended binary Golay code to be a perfect code (2008) Discrete Mathematics, 308, pp. 4057-4068Juba, B., Sudan, M., Universal semantics of Communication I - STOC'08 Proceedings of the 40th annual (2008) ACM symposium on Theory of computing, pp. 123-132Kim, H.K., Oh, D.Y., A Classification of Poset Admitting the MacWilliams Identity (2005) IEEE Transactions on Information Theory, 51 (4), pp. 1424-1431Kolmogorov, A.N., On the Shannon theory of information in the case of continuous signals (1956) IEEE Transactions on Information Theory, , SeptemberMasnick, B., Wolf, J., On linear unequal error protection codes (1967) IEEE Transactions on Information Theory, 3 (4), pp. 600-607Moura, A.D.O., Firer, M., Duality for poset codes (2010) IEEE Transactions on Information Theory, 56 (7), pp. 3180-3186Panek, L., Firer, M., Kim, H.K., Hyun, J.Y., Groups of linear isometries on poset structures (2008) Discrete Mathematics, 308, pp. 4116-4123Panek, L., Firer, M., Alves, M.M.S., Symmetry groups of Rosenbloom-Tsfasman spaces (2009) Discrete Mathematics, 309, pp. 763-771Panek, L., Firer, M., Alves, M.M.S., Classification of Niederreiter-Rosenbloom-Tsfasman Block Codes (2010) IEEE Transactions on Information Theory, 56 (10), pp. 5207-5216Parmigiani, G., Inoue, L.Y., Decision Theory, Principles and Approaches (2009) Wiley, Cornwall, , UKYu Rosenbloom, M., Tsfasman, M.A., Codes for the m-metric (1997) Problems of Information Transmission, 33, pp. 45-52Sason, I., Shamai, S., Performance Analysis of Linear Codes under Maximum-Likelihood Decoding (2006) A Tutorial - Foundations and Trends in Communication and Information Theory, 3 (1-2), pp. 1-222Shannon, C.E., A mathematical theory of communication (1948) Bell System Tech. J., 27, pp. 379-423Shannon, C.E., Coding theorems for a discrete source with a fidelity criterion (1959) IRE Nat. Conv. Rec., , marchSchröder, B.S.W., Ordered Sets - An Introduction (2003) BirkhäuserTavildar, S., Viswanath, P., Approximately universal codes over slow-fading channels (2006) IEEE Transactions on Information Theory, 52 (7), pp. 3233-3258XI Brazilian Meeting on Bayesian StatisticsSignificant part of this work was developed during the summer of 2011, when M. Firer was at the Centre Interfacultaire Bernoulli at EPFL, Lausanne, Switzerland, supported by the Swiss National Science Foundation, to which he deeply thanks. The second au

Linear codes on posets with extension property

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWe investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like. © 2013 Published by Elsevier B.V.We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like.3171113FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2007/56052-8; 2102/20181-7Aho, A., Hopcroft, J., Ullman, J.D., (1974) The Design and Analysis of Computer Algorithms, , Addison-Wesley Publishing Co. Reading, MAAlves, M.M.S., A standard form for generator matrices with respect to the Niederreiter-Rosenbloom-Tsfasman metric (2011) Proc. 2011 IEEE Information Theory Workshop, pp. 486-489. , 16-20 Oct. Paraty, BrazilBarg, A., Firer, M., Translation association schemes and shape enumerators of codes (2012) Proc. 2012 IEEE Internat. Sympos. Information Theory, pp. 101-105. , Boston, MA, JulyBarg, A., Purkayastha, P., Bounds on ordered codes and orthogonal arrays (2009) Mosc. Math. J., 9 (2), pp. 211-243Brouwer, A.E., Cohen, A.M., Neumaier, A., (1989) Distance-Regular Graphs, , Springer-Verlag Berlin e. aBrualdi, R.A., Graves, J.S., Lawrence, K.M., Codes with a poset metric (1995) Discrete Math., 147 (13), pp. 57-72Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F., (2008) Harmonic Analysis on Finite Groups, , Cambridge University PressChoi, S., Hyun, J.Y., Oh, D.Y., Kim, H.K., (2012) Mac-Williams Type Equivalence Relations, , arxiv:1205:1090Delsarte, P., An algebraic approach to the association schemes of coding theory (1973) Philips Res. Rep. Suppl., 10, pp. 1-97Delsarte, P., Association schemes and t-designs in regular semilattices (1976) J. Combin. Theory Ser. A, 20 (2), pp. 230-243Dougherty, S.T., Skriganov, M.M., MacWilliams duality and the Rosenbloom-Tsfasman metric (2002) Mosc. Math. J., 2 (1), pp. 81-97Felix, L.V., Firer, M., Canonical-systematic form for codes in hierarchical poset metrics (2012) Adv. Math. Commun., 6, pp. 315-328Gutiérrez, J.N., Tapia-Recillas, H., A MacWilliams identity for poset codes (1998) Congr. Numer., 133, pp. 63-73Hyun, J.Y., (2006) MacWilliams-type Equivalence Relations, , Ph.D. Thesis, Pohang University of Sciences and Technology, KoreaKim, D.S., Dual MacWilliams pairs (2005) IEEE Trans. Inform. Theory, 51 (8), pp. 2901-2905Kim, D.S., MacWilliams-type identities for fragment and sphere enumerators (2007) European Journal of Combinatorics, 28 (1), pp. 273-302. , DOI 10.1016/j.ejc.2005.07.018, PII S0195669805001526Kim, H.K., Oh, D.Y., A classification of posets-admitting the MacWilliams identity (2005) IEEE Transactions on Information Theory, 51 (4), pp. 1424-1431. , DOI 10.1109/TIT.2005.844067Lee, K., Automorphism group of the Rosenbloom-Tsfasman space (2003) European J. Combin., 24, pp. 607-612Lucas D'Oliveira, R.G., Firer, M., The Packing Radius of A Code and Partitioning Problems: The Case for Poset Metrics, , arxiv:1301.5915Macwilliams, F.J., Sloane, N.J.A., (1991) The Theory of Error-Correcting Codes, , North-Holland AmsterdamMartin, W.J., Stinson, D.R., Association schemes for ordered orthogonal arrays and (T, M, S) -nets (1999) Canad. J. Math., 51 (2), pp. 326-346Niederreiter, H., A combinatorial problem for vector spaces over finite fields (1991) Discrete Math., 96 (3), pp. 221-228Niederreiter, H., Digital nets and coding theory (2004) Coding Theory, Cryptography, and Combinatorics, pp. 247-257. , K. Feng, H. Niederreiter, C. Xing, Birkhäuser Basel e.aNielsen, R.R., A class of Sudan-decodable codes (2000) IEEE Trans. Inform. Theory, 46 (4), pp. 1564-1572Panek, L., Firer, M., Alves, M.M.S., Classification of Niederreiter-Rosenbloom-Tsfasman block codes (2010) IEEE Trans. Inform. Theory, 56 (10), pp. 5207-5216Panek, L., Firer, M., Kim, H.K., Hyun, J.Y., Groups of linear isometries on poset structures (2008) Discrete Math., 308 (18), pp. 4116-4123Pinheiro, J.A., Firer, M., Classification of poset-block spaces admitting MacWilliams-type identity (2012) IEEE Trans. Inform. Theory, 58 (12), pp. 7246-7252Reed, R.C., The coding of various kinds of unlabeled trees (1972) Graph Theory and Computing, pp. 153-182. , Academic Press New YorkRosenbloom, M.Yu., Tsfasman, M.A., Codes for the m-metric (1997) Probl. Inf. Transm., 33 (1), pp. 45-52Schmerl, J.H., Countable homogeneous partially ordered sets (1979) Algebra Universalis, 9, pp. 317-321Skriganov, M.M., Coding theory and uniform distributions (2001) Algebra i Analiz, 13 (2), pp. 191-239. , English translation in St. Petersburg Math. J. 13 (2) (2002) 301-337Stanley, R.P., (2012) Enumerative Combinatorics. Volume 1, , second ed. Cambridge University Press Cambridg