Coding Theory and Cryptography

final ctc logo 1 aug 31 2012

History and Description
The initiatives on coding theory and cryptography came to be in 2005 when Dr. Virgilio P. Sison completed his doctoral dissertation which dealt specifically with the construction of convolutional codes over the integer ring Z_{p^r} from linear block codes over the Galois ring GR(p^r,m). The distance of the block code provides a lower bound for the free distance of the convolutional code. The dissertation, which was supervised by Dr. Patrick Solé of CNRS, France, was published in the journal IEEE Transactions on Information Theory [3] and is now being cited as one of the construction methods for ring convolutional codes.

Dr. Sison has concentrated his research on the derivation of bounds for the distances of convolutional codes and codes over rings. On the side, new structural properties of finite rings, specifically Galois rings and finite Frobenius rings, are determined. Two other papers were published. The first, which generalizes Rabizzoni’s bound (1989), appeared in the IEEE Transactions on Information Theory [4], and is now being cited by MAGMA [5] as a reference for codes over Galois rings. The second, which generalizes Heller’s bound (1968), was presented in the International Conference in Mathematics and its Applications in Bangkok, Thailand in 2007, and was published in Matimyás Matematika, the official journal of the Mathematical Society of the Philippines [6]. Several other papers that resulted from Dr. Sison’s supervision of undergraduate and graduate researches were presented in local or international conferences and workshops. These papers are listed in the Research Outputs Section. Students and faculty members who were advised by Dr. Sison in their undergraduate and graduate researches are the members of the cluster.
At present, the cluster concentrates on the algebraic properties of block codes and convolutional codes over rings, the construction of convolutional codes from block codes, the derivation of certain distance bounds, and the structural properties of finite rings, particularly $F_{p^k} + u F_{p^k}$.

Objectives of the Cluster
The objectives of the cluster are the following:

  1. to help revitalize research in the Division by sustaining its cluster activities through long-term research projects with system funding, presentation of papers in conferences, publication in refereed journals, and sponsorship of seminars, conferences and schools on coding theory;
  2. to assist the Institute in its faculty development program by helping its members to obtain opportunities for trainings and workshops in coding theory and to complete their graduate research requirements;
  3. to support the curricular initiatives of the Division by proposing new courses on coding theory and cryptography; and finally,
  4. to foster collaboration among the different units of the University through inter-disciplinary researches.

Previous Activities
The cluster has conducted regular research discussions on the block distances of convolutional codes, and sponsored a lecture series on coding theory and a training workshop on the use of MAGMA. It has completed a basic research project titled Convolutional code constructions from block codes over Galois rings with Prof. Bernice Ruth C. Ilagan as Project Leader. Further it has aligned its special problem topics with the research thrusts of the cluster. The recently completed special problems dealt with constraint lengths of quaternary convolutional codes, multiplicative codes over the Gaussian units, compositions of Gray isometries, similar codes, homogeneous weights from generating characters of finite Frobenius rings, generalized orthogonal matrices over rings, codes over Galois ring extensions, codes and the hat game, generator matrices of cyclic codes, normal bases for Galois rings GR(4,m), matrix product codes, and with codes from linear block codes over GR(4,2). These special problem topics are listed in the Research Outputs Section.
Two members of the cluster, namely Prof. Palines and Prof. Lampos, have finished their graduate theses with topics in line with the research thrusts of the cluster.
During the last Math Consciousness Week, the cluster sponsored a popular lecture entitled “A Brief Introduction to Coding Theory.” The cluster invited Dr. Evangeline Bautista from Ateneo de Manila University to give the lecture.

Current Activities
The cluster intends to continue its regular research discussions and help its members to finish their graduate researches on the following topics:

  1. Bounds on the images of linear block codes over the finite chain rings $\FF_{p^r}+u\FF_{p^r}$ and their semi-local ring extensions, Ms. Jane D. Palacio, M.Sc. Mathematics
  2. Codes over finite Frobenius matrix rings, Mr. Dixie L. Falcunit, Jr., M.Sc. Mathematics

Cluster Members
Leader: Dr. Virgilio P. Sison
Members: Prof. Bernice Ruth C. Ilagan
Prof. John Mark T. Lampos
Prof. Herbert S. Palines
Mr. Bryan S. Hernandez
Ms. Jane D. Palacio
Mr. Dixie Falcunit (MS Student)
Mr. Charles R. Repizo (MS Alumnus)
The cluster counts as associate members the undergraduate students who are doing special problem researches in coding theory. It may invite affiliate members from other units in the University.

Future Plans
The cluster hopes to involve more students and faculty members in coding theory and cryptography. This may be addressed by the new graduate course (MATH 215 – Coding Theory and Cryptography) and MATH 200 (Undergraduate Thesis in Mathematics), the institution of which was initiated by the cluster. The course MATH 215 had six (6) students in its initial offering (first semester 2012-2013). The cluster also intends to explore more fields where coding theory and cryptography may be applied, such as digitization and DNA coding theory.

Research Outputs
Preprints
1. J.M. Lampos and V. Sison, Systematic unit-memory binary convolutional codes from linear block codes over $\FF_{2^r}+v\FF_{2^r}$, accepted in 2012 International Conference on Cryptography, Coding and Information Security (ICCCIS 2012), Venice, Italy, November 14-16, 2012
2. J.M. Lampos and V. Sison, Field codes from linear block codes over the semi-local ring $\FF_p+v\FF_p$ and their bounds, accepted in Matimyas Matematika, vol. 33, 2012
27 October 2012 Page 4
3. H. Palines and V. Sison, Deriving the parity check matrix of a ring convolutional code, accepted in Matimyas Matematika, vol. 33, 2012
4. H. Palines and V. Sison, A computer search algorithm for optiumum free distance binary convolutional codes, submitted to the 6th Symposium on the Mathematical Aspects of Computer Science (SMACS 2012), organized by the Computing Society of the Philippines, Boracay Island, Aklan, 4-8 December 2012
5. J. Palacio, V. Sison and J.M. Lampos, On the minimum distance of the p^m-ary image of linear block codes over the finite chain ring \FF_{p^m} + u\FF_{p^m} + u^2\FF_{p^m}+\ldots+u^{r-1} \FF_{p^m}, u^r=0, submitted to the 6th Symposium on the Mathematical Aspects of Computer Science (SMACS 2012), organized by the Computing Society of the Philippines, Boracay Island, Aklan, 4-8 December 2012
6. D. Falcunit, Jr. and V. Sison, Homogeneous weight and scaled isometry on \FF_{p^2}-linear map, submitted to the 6th Symposium on the Mathematical Aspects of Computer Science (SMACS 2012), organized by the Computing Society of the Philippines, Boracay Island, Aklan, 4--8 December 2012
7. V. Sison, Bases of the Galois ring GR(p^r,m) over the integer ring \ZZ_{p^r}, submitted to the Journal of Algebra and its Applications, August 2012
8. C. Repizo and V. Sison, One-sided $k$-orthogonal matrices over finite semi-local rings and their code-theoretic applications
9. V. Sison, New PUM code from a \ZZ_8-Kerdock code and RSC encoder from a quasi-cyclic $GR(4,2)$-code
10. V. Sison, The Singleton bound for convolutional codes over finite Frobenius rings

Papers Published in ISI international journals
1. P. Solé and V. Sison, “Bounds on the minimum homogeneous distance of the pr-ary image of linear block codes over the Galois ring GR(pr,m),” IEEE Transactions in Information Theory, vol. 53, no. 6, pp. 2270-2273, June 2007; also in the refereed Proceedings of the 2007 IEEE International Symposium on Information Theory (ISIT 2007), Acropolis Congress and Exhibition Center, Nice, France, pp. 1971-1974, 24-29 June 2007.
2. P. Solé and V. Sison, “Quaternary convolutional codes from linear block codes over Galois rings,” IEEE Transactions in Information Theory (ISIT 2007), vol. 53, no. 6, pp. 2267-2270, June 2007; also in the refereed Proceedings of the 2007 IEEE International Symposium on Information Theory, (ISIT 2007), Acropolis Congress and Exhibition Center, Nice, France, pp. 2641-2645, 24-29 June 2007.

Papers published in non-ISI refereed journals
1. J.M. Lampos and V. Sison, ``Bounds on the $p^k$-ary image of linear block codes over the semi-local Frobenius ring $\FF_{p^k}+v\FF_{p^k}$," accepted for publication in Southeast Asian J. of Sciences, 2012.
2. H. Palines and V. Sison, ``On self-dual convolutional codes over rings," accepted for publication in Southeast Asian J. of Sciences, 2012.
3. V. Sison, “Homogeneous bounds for the image of linear block codes over Galois ring extensions,” Matimyàs Matematika, Journal of the Mathematical Society of the Philippines, vol. 30, nos. 2-3, pp. 113-116, 2007.
27 October 2012 Page 5
4. V. Sison, “Heller-type bounds for the homogeneous free distance of convolutional codes over finite Frobenius rings,” Matimyàs Matematika, Journal of the Mathematical Society of the Philippines, vol. 30, no. 1, pp. 23-30, 2007; also in the Collection of Presented Papers of the International Conference in Mathematics and Applications (ICMA-MU 2007), The Century Park Hotel, Bangkok, Thailand, pp. 483-489, August 2007.
5. B.R. Cerezo, A.M. Pasion and V. Sison, “Binary convolutional codes and linear block codes over $\FF_4$,” Matimyàs Matematika, Journal of the Mathematical Society of the Philippines, vol. 29, no. 1-2, pp. 9-17, 2006; also in the peer-reviewed Pre-proceedings of the 3rd Symposium on the Mathematical Aspects of Computer Science} (SMACS 2006), pp. 77-86, October 2006.

Oral Presentations
International Level
1. J.M. Lampos and V. Sison, “Systematic Unit-Memory Binary Convolutional Codes from Linear Block Codes over $\FF_{2^r}+v\FF_{2^r}$,” accepted in 2012 International Conference on Cryptography, Coding and Information Security (ICCCIS 2012), Venice, Italy, 14-16 November 2012.
2. H. Palines and V. Sison, “On self-dual convolutional codes over the integer ring $\ZZ_{p^r}$,” International Conference in Mathematics and Applications (ICMA), University of Economics and Law, Vietnam National University, Ho Chi Minh City, Vietnam, 20-22 December 2011.
3. J.M. Lampos and V. Sison, “Bounds on the pk-ary image of linear block codes over the semi-local Frobenius ring $\FF_{p^k}+v\FF_{p^k}$,” International Conference in Mathematics and Applications (ICMA), University of Economics and Law, Vietnam National University, Ho Chi Minh City, Vietnam, 20-22 December 2011.
4. V. Sison, ``Convolutional codes over rings,'' Institut EURECOM, Sophia Antipolis, France, 7 December 2004; LACO-Université de Limoges, Limoges, France, 5 November 2004; CNRS-i3S-ESSI, Sophia Antipolis, France, 30 September 2004.

Master’s theses completed
1. “Distance bounds on the images of linear block codes over finite semi-local Frobenius rings,” John Mark T. Lampos, M.Sc. Mathematics, U.P. Los Baños, October 2011.
2. “Parity check matrices of convolutional codes over rings,” by Herbert S. Palines, M.Sc. Mathematics, U.P. Los Baños, June 2011
3. “Binary convolutional codes and linear block codes over $\FF_4$,” by Bernice Ruth M. Cerezo, M.Sc. Mathematics, U.P. Diliman, with Dr. Ann Marie Natividad-Pasion (U.P. Diliman), June 2006

Last Updated on Tuesday, 09 April 2013 03:37

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