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:

  1. •Gert Schneider (Ph.D. 2008): On the weight adjacency matrix of convolutional codes

  2. •Fai Lung Tsang (Ph.D. 2008): Skew rings, convolutional codes and discrete systems

  3. •Elizabeth Weaver (Ph.D. 2012): Minimality and duality of tail-biting trellises for linear codes

  4. •Aleams Barra (Ph.D. 2012): Equivalence theorems and the local-global property

  5. •Carolyn Troha (Ph.D. 2015): Analysis and Constructions of Subspace Codes

  6. •Neville Fogarty (Ph.D. 2016): On Skew-Constacyclic Codes

  7. •Isaiah Harney (Ph.D. expected 2017): Colorings of the Hamming graphs

  8. •Tefjol Pllaha (Ph.D. expected 2018): Extension theorems for sublinear isometries


Membership on Editorial Boards: 

  1. • Linear Algebra and its Applications (2016 - present)

  2. • SIAM Journal on Applied Algebra and Geometry (2016 - present)

  3. • Advances in Mathematics of Communications (2006 - present)

  4. • 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:

  1. •Journal Publications, Book Chapters, Preprints, and Theses:      

  2. ◦(with J. Antrobus) Lexicodes over Finite Principal Left Ideal Rings. Preprint 2016. Submitted. ArXiv:1612.04703

  3. ◦(with J. Bolkema, C.A. Kelley, K. Lauter, B. Malmskog, J. Rosenthal) Variations of the McEliece Cryptosystem. Preprint 2016. Submitted. ArXiv:1612.05085

  4. ◦(with T. Pllaha) Extension theorems for various weight functions over Frobenius bimodules. Preprint 2016. Submitted. ArXiv:1611.01141

  5. ◦(with I. Harney) On robust colorings of Hamming-distance graphs. Preprint 2016. Submitted. ArXiv:1609.00263

  6. ◦(with C. Troha) Construction of Subspace Codes Through Linkage. Advances in Mathematics of Communications 10 (2016),  525 - 540.

  7. ◦(with N. Fogarty) A Circulant Approach to Skew-Constacyclic Codes. Finite Fields and their Applications 35 (2015),  92 - 114.

  8. ◦The Homogeneous Weight Partition and its Character-Theoretic Dual. Designs, Codes and Cryptography 79 (2016), 47 -  61.

  9. ◦(with K. Morrison and C. Troha) Cyclic Orbit Codes and Stabilizer Subfields. Advances in Mathematics of Communications 9 (2015), 177 - 197.

  10. ◦(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.

  11. ◦Partitions of Frobenius Rings Induced by the Homogeneous Weight. Advances in Mathematics of Communications, 8 (2014),       191 - 207.

  12. ◦ Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes. Designs, Codes and Cryptography 75 (2015), 543 -   563.

  13. ◦     (with G.D. Forney, Jr) Local Irreducibility of Tail-Biting Trellises IEEE Trans. Inf. Theory 59 (2013), 6597-6610.

  14. ◦     (with G.D. Forney, Jr) Codes on Graphs: Observability, Controllability and Local Irreducibility, IEEE Trans. Inf. Theory 59 (2013),

                      223 - 237

    1. ◦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.

    2. ◦(with E. Weaver) Characteristic generators and Dualization of Tail-Biting Trellises, IEEE Trans. Inf. Theory 57 (2011), 7418 - 7430

    3. ◦(with E. Weaver) Linear Tail-Biting Trellises: Characteristic generators and the BCJR-Construction, IEEE Trans. Inf. Theory 57 (2011), 738-751

    4. ◦(with U. Helmke and J. I. Iglesias Curto) Algebraic Decoding for Doubly Cyclic Convolutional Codes, Advances in Mathematics of Communications 4(2010), 83-99

    5. ◦(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

    6. ◦On isometries for convolutional codes, Advances in Mathematics of Communications 3 (2009), 179-203

    7. ◦(with G. Schneider) A MacWilliams Identity for convolutional codes: The general case, IEEE Trans. Inf. Theory 55 (2009), 2920-2930

    8. ◦(with F.-L. Tsang) A matrix ring description for cyclic convolutional codes, Advances in Mathematics of Communications 2 (2008), 55-81

    9. ◦(with G. Schneider) State space realizations and monomial equivalence for convolutional codes, Linear Algebra and its Applications 425 (2007), 518-533

    10. ◦(with G. Schneider) On the MacWilliams Identity for convolutional codes, IEEE Trans. Inf. Theory 54 (2008), 1536-1550

    11. ◦On the weight distribution of convolutional codes. Linear Algebra and its Applications 408 (2005), 298-326.

    12. ◦(with B. Langfeld) A class of one-dimensional MDS convolutional codes. Journal of Algebra and its Applications 5 (2006), 505-520.

    13. ◦(with W. Schmale) On doubly-cyclic convolutional codes. Applicable Algebra in Engineering, Communication and Computing 17 (2006), 151-170.

    14. ◦(with B. Langfeld) On the algebraic parameters of convolutional codes with cyclic structures Journal of Algebra and its Applications 5 (2006), 53-76.

    15. ◦(with W. Schmale) Distance bounds for convolutional codes and some optimal codes, 2003. 22 pages.

    16. ◦(with R. Smarandache und J. Rosenthal) Strongly MDS convolutional codes, IEEE Trans. Inf. Theory 52 (2006), 584-598

    17. ◦(with W. Schmale) On cyclic convolutional codes, Acta Applicandae Mathematicae 82 (2004), 183-237

    18. ◦(with R. Smarandache und J. Rosenthal) Constructions of MDS-Convolutional Codes, IEEE Trans. Inf. Theory 47 (2001), 2045-2049

    19. ◦(with S. Zampieri und P. Vettori) The algebraic structure of DD systems: a behavioral perspective, Kybernetika 37 (2001), 397-426

    20. ◦(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

    21. ◦The Fuhrmann-realization for multi-operator systems in the behavioral context , Multidimensional Systems and Signal Processing 11 (2000), 193-211

    22. ◦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

    23. ◦First-order representations of delay-differential systems in a behavioural setting, European Journal of Control 3 (1997), 137-149

    24. ◦A behavioral approach to delay-differential systems, SIAM Journal on Control and Optimization 35 (1997), 480-499

    25. ◦Continuous state-representations for AR-systems, Mathematics of Control, Signals, and Systems 8 (1995), 82-95

    26. ◦Realization of rational matrices by singular systems, Journal of Mathematical Systems, Estimation, and Control 5 (1995), no. 1, 1-32

    27. ◦(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

    28. ◦(with D. Hinrichsen) The degeneration of reachable singular systems under feedback-transformations, Kybernetika 30 (1994), 387-391

    29. ◦On various topologies for finite-dimensional linear systems, Institute for Dynamical Systems, University of Bremen, Report No. 273, December 1992, 39 pages

    30. ◦Gruppenaktionen in der Theorie singulärer Systeme, Dissertation, University of Bremen, 1991

    31. ◦Feedback canonical form for singular systems, International Journal of Control 52 (1990), 347--376

    32. ◦(with D. Hinrichsen) A Jordan control canonical form for singular systems, International Journal of Control 48 (1988), 1769-1785

  2. •Monograph:

    1. ◦Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach; Lecture Notes in Mathematics 1770; Springer 2002.