## Research

### Journal papers and book chapters

Please, fell free of asking for reprints or electronic files of them to me. If you have access to AMS MathScinet you can find my publication profile in EMM's AMS profile.

Galois LCD codes over mixed alphabets
(with
Maryam Bajalan, Alexandre Fotue and Joel Kabore)
Finite Fields and Their Applications ** Open access** under a Creative Commons license. (D.O.I. 10.1016/j.ffa.2022.102125)

Public Key Protocols over Skew Dihedral Group Rings (with
Javier de la Cruz and Ricardo Villanueva Polanco)
Mathematics (2022) ** Open access** (D.O.I. 10.3390/math10183343)

Free Resolutions and Generalized Hamming Weights of Binary Linear Codes (with
Ignacio Marco, Irene Marquez and Yuriko Pitones)
Mathematics (2022) ** Open access** (D.O.I. 10.3390/math10122079)

Lifted codes and lattices from codes over finite chain rings (with Lamia Bouzara and Kenza Guenda) Cryptography and Communications (2022) (D.O.I. 10.1007/s12095-022-00568-5)

A transform approach to polycyclic and serial codes over rings (with
Maryam Bajalan and Steve Szabo)
Finite Fields and Their Applications (2022) ** Open access** under a Creative Commons license. (D.O.I. 10.1016/j.ffa.2022.102014)

Hulls of cyclic serial codes over a finite chain ring (with
Sarra Talbi, Aicha Batoul and Alexandre Fotue-Tabue)
Finite Fields and Their Applications (2022)** Open access** under a Creative Commons license.
(D.O.I. 10.1016/j.ffa.2021.101950)

Cyclic codes over a non-chain ring Re,q and their application to LCD codes (with Habibul Islam and Om Prakash) Discrete Mathematics (2021) (D.O.I. 10.1016/j.disc.2021.112545)

Linear complementary pair of group codes over finite chain rings (with Cem Güneri and Selcen Sayici) Designs, Codes and Cryptography (2020) (D.O.I. 10.1007/s10623-020-00792-1)

Computing sharp recovery structures for locally recoverable codes (with I. Marquez-Corbella and C. Munuera) Designs, Codes and Cryptography (2020) (D.O.I. 10.1007/s10623-020-00746-7)

Do non-free LCD codes over finite commutative Frobenius rings exist? (with S. Bhowmick, A. Fotue-Tabue, R. Bandi, S. Bagchi) Designs, Codes and Cryptography (2020) (D.O.I. 10.1007/s10623-019-00713-x)

Galois correspondence on linear codes over finite chain rings (with A. Fotue and C. Mouaha) Discrete Mathematics (2020) (D.O.I. 10.1016/j.disc.2019.111653)

On polycyclic codes over a finite chain ring (with A. Fotue and J.T. Blackford) Advances in Mathematics of Communications (2020) (D.O.I. 10.3934/amc.2020028)

Vardøhus Codes: Polar Codes Based on Castle Curves Kernels (with E. Camps and E. Sarmiento) IEEE Transactions on Information Theory (2020) (D.O.I. 10.1109/TIT.2019.2932405)

General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings (with A. Cuida, F. Laudano) International Journal of Mathematical Education in Science and Technology (2019) (D.O.I. 10.1080/0020739X.2019.1676926)

On counting subring-submodules of free modules over finite commutative Frobenius rings (with R. Bandi and A. Fotue) Designs, Codes and Cryptography (2018) (D.O.I. 10.1007/s10623-017-0446-1)

On the Weak Order Ideal Associated to Linear Codes. (With Mijail Borges-Quintana, Miguel A. Borges-Trenard) Mathematics in Computer Science (2018) (D.O.I. 10.1007/s11786-018-0349-1)

Additive cyclic codes over finite commutative chain rings (with K. Otal and Ferruh Ozbudak) Discrete Mathematics (2018) (D.O.I. 10.1016/j.disc.2018.03.016)

Codes over affine algebras with a finite commutative chain coefficient ring (with I. F. Rua and A. P. Nicolas) Finite Fields and Their Applications (2018) (D.O.I. 10.1016/j.ffa.2017.09.008)

Multivariable codes in principal ideal polynomial quotient rings with applications to additive modular bivariate codes over F4 (with I. F. Rua and A. P. Nicolas) Journal of Pure and Applied Algebra (2018) (D.O.I. 10.1016/j.jpaa.2017.04.007)

Trial Set and Groebner Bases for Binary Codes. (With Mijail Borges-Quintana, Miguel A. Borges-Trenard) Springer Proceedings in Mathematics & Statistics (2017).

Applications of Computer Algebra . (Edited with Ilias Kotsireas), Springer Proceedings in Mathematics & Statistics (2017). Foreword - Front matter

On the ideal associated to any linear code (with I. Marquez-Corbella and E. Suarez) Advances in Mathematics of Communications (2016) (D.O.I. 10.3934/amc.2016003)

Algebra for Secure and Reliable Communication Modeling . (Edited with Mustapha Lahyane) , Contemp. Math., Amer. Math. Soc (2015). (DOI:10.1090/conm/642). Foreword - Front matter

A Semigroup Approach to Complete Decoding . (With I. Marquez-Corbella) Algebra for Secure and Reliable Communication Modeling, Contemp. Math., Amer. Math. Soc (2015).

**Computer algebra in coding theory and cryptography** (Edited with Ilias S. Kotsireas)
Special Issue of
Designs, Codes and Cryptography.
July 2015, Volume 76, Issue 1.

Computing Coset Leaders and Leader Codewords of Binary Codes. (With M.A. Borges, M. Borges and I. Marquez-Corbella) Journal of Algebra and its Applications, Volume 14, Issue 08, (2015).(DOI: 10.1142/S0219498815501285)

On a class of repeated-root monomial-like abelian codes. (With H. Ozadan, F. Ozbudak and S. Szabo) Journal of Algebra Combinatorics Discrete Structures and Applications, Volume: 2, Issue: 2, pp:1-10 (2015).

Linear Codes Over Z_4 [x]/(x^2 +2x). (With S. Szabo and B. Yildiz) The International Journal of Information and Coding Theory, Vol.3, No.1 pp.78 - 96 (2015).

On codes over local Frobenius non-chain rings of order 16. (With S. Szabo) In Noncommutative Rings and Their Applications, volume 634 of Contemp. Math., pages 227-243. Amer. Math. Soc (2015).

On the Composition of Secret Sharing Schemes Related to codes. (With I. Marquez-Corbella and E. Suarez-Canedo) Discrete Math., Alg. and Appl. 6(1) (2014). (DOI: 10.1142/S179383091450013X)

On additive modular bivariate codes over F_4. (With A.P. Nicolas and I.F. Rua) Finite Fields and their Applications. 28, pp. 199 - 213. (2014). (DOI: 10.1016/j.ffa.2014.02.007)

Computational aspects of retrieving a representation of an algebraic geometry code. (with I. Marquez-Corbella, R. Pellikaan and D. Ruano) Journal of Symbolic Computation (2014). (DOI: 10.1016/j.jsc.2013.12.007)

**Computer algebra in coding theory and cryptography** (Edited with Iliya Bouyukliev, Stanislav Bulygin)
Special Issue of
Applicable Algebra in Engineering, Communication and Computing.
August 2013, Volume 24, Issue 3-4.

On Trace Codes and Galois Invariance over Finite Commutative Chain Rings. (with A.P. Nicolas and I.F. Rua) Finite Fields and Their Applications Volume 22, July 2013, Pages 114-121. (DOI: 10.1016/j.ffa.2013.03.004) Corrigendum (DOI: 10.1016/j.ffa.2013.05.002)

An Introduction to LDPC Codes. (with I. Marquez-Corbella) World Scientific Series on Coding Theory and Cryptology: Volume 8.

On the unique representation of very strong algebraic geometry codes. (with I. Marquez-Corbella and R. Pellikaan) Designs, Codes and Cryptography. (2013). (DOI: 10.1007/s10623-012-9758-3)

The non-gap sequence of a subcode of a generalized Reed-Solomon code. (with I. Marquez-Corbella and R. Pellikaan) Designs, Codes and Cryptography. (2013). (DOI: 10.1007/s10623-012-9694-2)

Additive semisimple multivariable codes over F_4. (with I. F. Rua and A. P. Nicolas) Designs, Codes and Cryptography.Volume 69, Issue 2, pp 161-180 (2013). (DOI: 10.1007/s10623-012-9641-2)

**Matroids in Coding Theory and Related Topics** (Edited with Ilias S. Kotsireas and Irene Marquez-Corbella)
Special Issue of
Mathematics in Computer Science. Volume 6, Number 2 (2012).

Decomposition of Modular Codes for Computing Test Sets and Graver Basis . (with I. Marquez-Corbella) Mathematics in Computer Science, volume 6, Number 2, (2012). Pages: 147-165 (DOI: 10.1007/s11786-012-0120-y)

Algebraic structure of the minimal support codewords set of some linear codes . (with I. Marquez-Corbella) Advances in Mathematics of Communications (AMC), volume 5, Issue 2, May 2011. Pages: 233-244 (DOI: 10.3934/amc.2011.5.233)

**An Algebraic View to Gradient Descent Decoding** (with E. Martinez-Moro, M. ángel Borges Trenard, Mijail
Borges Quintana)
IEEE Information Theory Workshop, Dublin, August 30 - September 3, 2010. IEEExplore

**Algebraic Coding Theory
and Applications** (Edited with Antonio Campillo, Patrick Fitzpatrick and Ruud Pellikaan)
Special Issue of the
Journal of Symbolic Computation. Volume 45, Issue 7, Pages 721-824 (July 2010)

Gröbner representations of binary matroids (With M. Borges-Quintana, M. A. Borges-Trenard) In Bras-Amoros, Maria (ed.) et al., Applied algebra, algebraic algorithms and error-correcting codes. 18th international symposium, AAECC-18. Lecture Notes in Computer Science 5527, 227-230 (2009).

**An application of Möller's algorithm to coding theory** (With M. Borges-Quintana, M. A. Borges-Trenard) In
Gröbner bases, coding, and cryptography. 379-384
Edited by Massimiliano Sala, Teo Mora, Ludovic Perret, Shojiro Sakata and Carlo Traverso. Springer-Verlag, Berlin, 2009.

**Mattson Solomon transform and algebra codes** (With D. Ruano). In Gröbner bases, coding, and cryptography. 385-388
Edited by Massimiliano Sala, Teo Mora, Ludovic Perret, Shojiro Sakata and Carlo Traverso. Springer-Verlag, Berlin, 2009.

**Toric codes** (With D. Ruano). In Advances in Algebraic
Geometry Codes, 295-322, Ser. Coding Theory Cryptol., 5, World Sci. Publ., Hackensack, NJ, 2008.

Gröbner bases and combinatorics for binary codes (with M. Borges Quintana, M.A. Borges-Trenard and P. Fitzpatrick) in Applicable Algebra in Engineering, Communication and Computing. (DOI: 10.1007/s00200-008-0080-2)

On semisimple algebra codes: generator theory
Algebra and Discrete Mathematics.(2007) Number 3.

On repeated-root multivariable codes over a finite chain ring
(with I.F. Rua) Designs, Codes and
Cryptography 45:2 November 2007.(DOI: 10.1007/s10623-007-9114-1)

**A Gröbner Representation for Linear Codes **. (with M. Borges Quintana and M.A. Borges-Trenard) In
Advances in Coding Theory and Cryptography
edited by T Shaska, W C Huffman, D Joyner and V Ustimenko. Pages 17-32. Aug. 2007. World Scientific.

**On a Gröbner bases structure associated to linear codes**. (with M. Borges
Quintana and M.A. Borges-Trenard) Journal of Discrete Mathematical Sciences and Cryptography. March 2007. 151-191

A General Framework for Applying FGLM Techniques to Linear Codes
. (with M. Borges-Quintana and M.A. Borges-Trenard) AAECC'16, Lecture Notes in Computer Science, (3857). Dec. 2006.
Pages: 76-86 (DOI: 10.1007/11617983_7)

Multivariable Codes Over Finite Chain Rings: Serial Codes. (with
I.F. Rua) SIAM Journal on Discrete Mathematics. Volume 20 Issue 4 Dec. 2006. Pages 947-959. (DOI: 10.1137/050632208)

The symmetric group given by a Gröbner basis. (with M. Borges-Quintana, M.A. Borges-Trenard and J.A. Castellanos-Garzón) Journal of Pure and Applied Algebra
Volume 207, Issue 1, September 2006, Pages 149-154. (DOI: 10.1016/j.jpaa.2005.09.011)

Compounding Secret Sharing Schemes.
(with C.Munuera and J. Mozo) Australian Journal of Combinatorics, 30, September 2004, pages 277-290.

A generalization of Niederreiter-Xing's propagation
rule and its commutativity with duality. IEEE Trans. on Inf. Theory, Volume: 50,Issue: 4 April 2004, pp. 701- 702.(DOI: 10.1109/TIT.2004.825036)

Regular Representations of Finite-dimensional Separable Semisimple Algebras and Gröbner Bases. Jour.
Symbolic Computation. Volume 37, Issue 5, May 2004, Pages 575-587

(DOI: 10.1016/j.jsc.2003.08.001)

On the structure of multipliers
of Z^{2}. (with R. Canogar-McKenzie) . Int. Jour. Mathematics and Mathematical Sciences. vol. 15 (2003), Pages: 935--946
(DOI: 10.1155/S0161171203203124)

Properties of Commutative Association
Schemes derived by FGLM Techniques.International Journal of Algebra and Computation. vol. 12 no. 6 (2002). Pages: 849--865
(DOI: 10.1142/S021819670200119X)

Majority-logic-decodable
cyclic arithmetic-modular AN-codes in 1, 2 and L steps. (with F. Javier Galán y J.G. Tena). In Cryptography and Coding. Bahram Honary (Ed.).
Lecture Notes in Computer Science. Springer Verlag. Vol. 2260, Páginas: 128-137, 2001

Combinatorial
Structure of Finite Fields with Two Dimensional Modulo Metrics. (with F. Javier Galán Simon, M. ángel Borges Trenard, Mijail
Borges Quintana). In Cryptography and Coding.Mike Walker (Ed.). Lecture Notes in Computer Science. Springer Verlag. Vol. 1746, Pages: 45-55, 1999.

**Computations on character tables of association schemes**. In Computer Algebra in Scientific Computing. CASC99.
V.G. Ganzha, E.W. Mayr, E.W. Vorozhtsov (Eds.) Springer Verlag, Munich 1999. Pages: 293-307

**Exploring self-dual codes with Maple**. Maple Tech. Birhäuser. Volumen: 5, 1. Pages 37-41. 1998.

### Books

Algebraic Geometry Modeling in Information Theory. (Editor) Feb 2013. World Scientific Series on Coding and Cryptography, vol. 8.

Advances in Algebraic Geometry Codes (Editor with Diego Ruano y Carlos Munuera). May 2008. World Scientific Series on Coding and Cryptography. vol 5.

Introducción a los anillos finitos y sus aplicaciones (in Spanish)
con A. P. Nicolas y I.F. Rua. Septiembre 2014. Instituto Venezolano de Investigaciones Científicas. 121 páginas.
(ISBN: 78-980-261-151-5)

Bases de Gröbner: aplicaciones
a la codificación algebraica (in Spanish) con Diego Ruano y Carlos Munuera. Septiembre 2007. Instituto Venezolano
de Investigaciones Científicas. 125 páginas.
(ISBN: 978-980-261-087-7)

V Jornadas de Matemática Discreta y Algorítmica.
(Coordinador,in Spanish) Julio 2006. Secretariado de Publicaciones de la UVa. 446 páginas. (ISBN: 84-8448-380-0)

### Software

New decoding
error-correcting codes algorithm for Sage
by Verónica Suaste, supported by Google Summer of code 2013.

GBLA_LC: A GAP package for computing Grobner Bases by Linear. Algebra and Linear Codes.

Presented at the International Congress of Mathematics (with M. Borges-Quintana and M.A. Borges-Trenard), see the
extended abstract.

### Supervised students

**Cristina Garcia Gonzalez**
Undergraduate. Graduate Thesis. Main results publised as:

Quintana, M. B.; Trenard, M. B.; Garcia-Gonzalez, C.; Martinez, E. **The partial order associated with the equivalence classes of a binary code**. (Spanish)
Sixth Conference on Discrete Mathematics and Computer Science (Spanish), 509-516, Univ. Lleida, Lleida, 2008.

**Irene Marquez Corbella** PhD Student. FPU Grant form Spanish Research Ministery. September
2009-2013.

Master
Tesis
Presentaciones binomiales de programas lineales
modulares: Aplicaciones a los códigos correctores de errores
(in Spanish) 07/2010. Calificación: Sobresaliente (9,5).

PhD Thesis: July 2013 Combinatorial Commutative Algebra Approach to Complete Decoding. Sobresaliente Cum Laude.

### PhD Thesis

I quite doubt it but maybe someone is interested in my PhD thesis: Estructura combinatoria
de métricas no Hamming (in Spanish) 07/2001. Advisor: Dr. Fco. Javier Galán Simón.
Calificación: Sobresaliente "cum laude".

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