Research
Research
My current research interests are in algebraic coding theory with special interest in trellis representations of linear block
codes, subspace codes and MacWilliams identities and extension theorems. I also worked in algebraic systems theory.
I am a member of the algebra group. We run an almost weekly seminar.
Ph.D. Students:
•Gert Schneider (Ph.D. 2008): On the weight adjacency matrix of convolutional codes
•Fai Lung Tsang (Ph.D. 2008): Skew rings, convolutional codes and discrete systems
•Elizabeth Weaver (Ph.D. 2012): Minimality and duality of tail-biting trellises for linear codes
•Aleams Barra (Ph.D. 2012): Equivalence theorems and the local-global property
•Carolyn Troha (Ph.D. 2015): Analysis and Constructions of Subspace Codes
•Neville Fogarty (Ph.D. 2016): On Skew-Constacyclic Codes
•Isaiah Harney (Ph.D. expected 2017): Colorings of the Hamming graphs
•Tefjol Pllaha (Ph.D. expected 2018): Extension theorems for sublinear isometries
Membership on Editorial Boards:
• Linear Algebra and its Applications (2016 - present)
• SIAM Journal on Applied Algebra and Geometry (2016 - present)
• Advances in Mathematics of Communications (2006 - present)
• SIAM Journal on Control and Optimization (2003 - 2014)
External Funding:
NSF Grant (Oct 1, 2009 - Sep 30, 2012): $183,418
NSF Grant (Sep 15, 2012 - Aug 31, 2015): $168,033
Simons Foundation Collaboration Grant (Sep 1, 2016 - Aug 31, 2021): $35,000
Publications:
•Journal Publications, Book Chapters, Preprints, and Theses:
◦(with J. Antrobus) Lexicodes over Finite Principal Left Ideal Rings. Preprint 2016. Submitted. ArXiv:1612.04703
◦(with J. Bolkema, C.A. Kelley,
K. Lauter, B. Malmskog, J. Rosenthal) Variations of the McEliece Cryptosystem. Preprint 2016. Submitted. ArXiv:1612.05085
◦(with T. Pllaha) Extension theorems for various weight functions over Frobenius bimodules. Preprint 2016. Submitted. ArXiv:1611.01141
◦(with I. Harney) On robust colorings of Hamming-distance graphs. Preprint 2016. Submitted. ArXiv:1609.00263
◦(with C. Troha) Construction of Subspace Codes Through Linkage. Advances in Mathematics of Communications 10 (2016), 525 - 540.
◦(with N. Fogarty) A Circulant Approach to Skew-Constacyclic Codes. Finite Fields and their Applications 35 (2015), 92 - 114.
◦The Homogeneous Weight Partition and its Character-Theoretic Dual. Designs, Codes and Cryptography 79 (2016), 47 - 61.
◦(with K. Morrison and C. Troha) Cyclic Orbit Codes and Stabilizer Subfields. Advances in Mathematics of Communications 9 (2015), 177 - 197.
◦(with A. Barra) MacWilliams Extension Theorems and the Local-Global Property for Codes over Frobenius Rings. Journal of Pure and Applied Algebra 219 (2015), 703 - 728.
◦Partitions of Frobenius Rings Induced by the Homogeneous Weight. Advances in Mathematics of Communications, 8 (2014), 191 - 207.
◦ Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes. Designs, Codes and Cryptography 75 (2015), 543 - 563.
◦ (with G.D. Forney, Jr) Local Irreducibility of Tail-Biting Trellises IEEE Trans. Inf. Theory 59 (2013), 6597-6610.
◦ (with G.D. Forney, Jr) Codes on Graphs: Observability, Controllability and Local Irreducibility, IEEE Trans. Inf. Theory 59 (2013),
223 - 237
◦On MacWilliams Identities for Codes Over Rings, in: K. Hüper and J. Trumpf (Ed.), Mathematical System Theory - Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, CreateSpace, 2013.
◦(with E. Weaver) Characteristic generators and Dualization of Tail-Biting Trellises, IEEE Trans. Inf. Theory 57 (2011), 7418 - 7430
◦(with E. Weaver) Linear Tail-Biting Trellises: Characteristic generators and the BCJR-Construction, IEEE Trans. Inf. Theory 57 (2011), 738-751
◦(with U. Helmke and J. I. Iglesias Curto) Algebraic Decoding for Doubly Cyclic Convolutional Codes, Advances in Mathematics of Communications 4(2010), 83-99
◦(with B. Langfeld and W. Schmale) A Short Introduction to Cyclic Convolutional Codes, in: Sala, M.; Mora, T.; Perret, L.; Sakata, S.; Traverso, C. (Ed.), Gröbner Bases, Coding, and Cryptography, RISC Book Series, Springer, Heidelberg (2009), 403-408
◦On isometries for convolutional codes, Advances in Mathematics of Communications 3 (2009), 179-203
◦(with G. Schneider) A MacWilliams Identity for convolutional codes: The general case, IEEE Trans. Inf. Theory 55 (2009), 2920-2930
◦(with F.-L. Tsang) A matrix ring description for cyclic convolutional codes, Advances in Mathematics of Communications 2 (2008), 55-81
◦(with G. Schneider) State space realizations and monomial equivalence for convolutional codes, Linear Algebra and its Applications 425 (2007), 518-533
◦(with G. Schneider) On the MacWilliams Identity for convolutional codes, IEEE Trans. Inf. Theory 54 (2008), 1536-1550
◦On the weight distribution of convolutional codes. Linear Algebra and its Applications 408 (2005), 298-326.
◦(with B. Langfeld) A class of one-dimensional MDS convolutional codes. Journal of Algebra and its Applications 5 (2006), 505-520.
◦(with W. Schmale) On doubly-cyclic convolutional codes. Applicable Algebra in Engineering, Communication and Computing 17 (2006), 151-170.
◦(with B. Langfeld) On the algebraic parameters of convolutional codes with cyclic structures Journal of Algebra and its Applications 5 (2006), 53-76.
◦(with W. Schmale) Distance bounds for convolutional codes and some optimal codes, 2003. 22 pages.
◦(with R. Smarandache und J. Rosenthal) Strongly MDS convolutional codes, IEEE Trans. Inf. Theory 52 (2006), 584-598
◦(with W. Schmale) On cyclic convolutional codes, Acta Applicandae Mathematicae 82 (2004), 183-237
◦(with R. Smarandache und J. Rosenthal) Constructions of MDS-Convolutional Codes, IEEE Trans. Inf. Theory 47 (2001), 2045-2049
◦(with S. Zampieri und P. Vettori) The algebraic structure of DD systems: a behavioral perspective, Kybernetika 37 (2001), 397-426
◦(with J. Rosenthal und P. Weiner) Duality between Multidimensional Convolutional Codes and Systems, in: F. Colonius, U. Helmke, F. Wirth, D. Praetzel-Wolters (Ed.), Advances in Mathematical Systems Theory, Birkhauser Boston, 2000, 135-150
◦The Fuhrmann-realization for multi-operator systems in the behavioral context , Multidimensional Systems and Signal Processing 11 (2000), 193-211
◦A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability, Mathematics of Control, Signals, and Systems 13 (2000), 22-40
◦First-order representations of delay-differential systems in a behavioural setting, European Journal of Control 3 (1997), 137-149
◦A behavioral approach to delay-differential systems, SIAM Journal on Control and Optimization 35 (1997), 480-499
◦Continuous state-representations for AR-systems, Mathematics of Control, Signals, and Systems 8 (1995), 82-95
◦Realization of rational matrices by singular systems, Journal of Mathematical Systems, Estimation, and Control 5 (1995), no. 1, 1-32
◦(with F. De Mari) The spaces of improper rational matrices and ARMA-systems of fixed McMillan degree, Journal of Mathematical Systems, Estimation, and Control 5 (1995), no. 1, 1-19
◦(with D. Hinrichsen) The degeneration of reachable singular systems under feedback-transformations, Kybernetika 30 (1994), 387-391
◦On various topologies for finite-dimensional linear systems, Institute for Dynamical Systems, University of Bremen, Report No. 273, December 1992, 39 pages
◦Gruppenaktionen in der Theorie singulärer Systeme, Dissertation, University of Bremen, 1991
◦Feedback canonical form for singular systems, International Journal of Control 52 (1990), 347--376
◦(with D. Hinrichsen) A Jordan control canonical form for singular systems, International Journal of Control 48 (1988), 1769-1785
•Monograph:
◦Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach; Lecture Notes in Mathematics 1770; Springer 2002.