Workshop on Algebraic Coding Theory And Related Concepts
Program and Abstracts
Morning 25/12/2019:
8:00 – 8:30 Registration
8:30 – 8:40 Open Ceremony
8:40 – 10:00 Hai Dinh, Kent State University and VIASM
Lecture 1 - Introduction to algebraic coding theory: the choice of alphabets.
Lecture 2 - Basic concepts in codes and rings.
Lecture 3 - Cyclic codes over finite commutative rings: the simple-root case.
10:00 – 10:30 Tea break
10:30 – 11:50 Hai Dinh, Kent State University and VIASM
Continue Lecture 1, 2, 3
Afternoon 25/12/2019:
13:30 – 14:50 Hai Dinh, Kent State University and VIASM
Lecture 4 - Galois extension rings of F2+uF2 and their use as code-alphabets.
Lecture 5 - Repeated-root contacyclic codes of prime power length over finite chain rings.
Lecture 6 - Repeated-root constacyclic codes of length p^s over F+uF.
14:50 – 15:20 Tea break
15:20 – 16:40 Hai Dinh, Kent State University and VIASM
Continue Lecture 4, 5, 6
Morning 26/12/2019:
8:30 – 9:50 Hai Dinh, Kent State University and VIASM
Lecture 7 - Repeated-root constacyclic codes of length 2p^s and 4p^s over F+uF.
Lecture 8 - Skew constacyclic codes over finite fields and finite chain rings.
9:00 – 10:20 Tea break
10:20 – 11:40 Hai Dinh, Kent State University and VIASM
Continue Lecture 7, 8
Afternoon 26/12/2019:
14:00 – 14:50 Truong Cong Quynh, Danang University and VIASM
Modules close to the automorphism-invariant and coinvariant
14:50 – 15:05 Tran Hoai Ngoc Nhan, Vinh Long University
Some characterizations of A-C3 modules
15:05 – 15:20: Banh Duc Dung, Ho Chi Minh City University of Technology and Education
A note on Harada rings
15:20 – 15:50 Tea break
15:50 – 16:05 Nguyen Thi Thu Ha, Industrial University of Ho Chi Minh City
On essentially injective modules
16:05 – 16:20 Phan Dan, Hong Bang International University
Characterizations of rings using quasi-injective modules
16:20 – 16:35 Le Van Thuyet, Hue University
A note on ring whose maximal right ideals are finitely generated
16:35 – 16:50 Dao Thi Trang, Ho Chi Minh City University of Food Industry
A note on automorphism-invariant modules
Morning 27/12/2019: Abhay Kumar Singh, Indian Institute of Technology Dhanbad, India
8:30 – 9:50: Overview on Code-based cryptography
9:00 – 10:20 Tea break
10:20 – 11:40 Continue
Afternoon 27/12/2019: Abhay Kumar Singh, Indian Institute of Technology Dhanbad, India
13:30 – 14: DNA Cyclic Codes of Finite Rings
14:50 – 15:20 Tea break
15:20 – 16:40 Continue
16:40 – 17:00 Closing ceremony
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ABSTRACTS:
Lecture Series on Coding Theory
Hai Dinh
Kent State University and VIASM
The existence of noise in communication channels is an unavoidable fact of life. The common feature of communication channels is that the original information is sent across a noisy channel to a receiver at the other end, where the received message is not always the same as what was sent. The fundamental problem is to detect if there is an error, and in such case, to determine what message was sent based on the approximation that was received. A response to this problem has been the creation of error-correcting codes. Coding Theory has grown into an important subject that intersects various scientific disciplines, such as information theory, electrical engineering, mathematics, and computer science, for the purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the detection and correction of errors in the transmitted data. In this lecture series, we will discuss the historical background of Algebraic Coding Theory, and how new researchers can start doing research in Coding Theory by studying the algebraic structure of the so-called constacyclic codes. Foundational and theoretical aspects of the role of finite rings as alphabets in coding theory are discussed, with a concentration on the class of constacyclic codes over finite commutative chain rings. We study both the simple-root and repeated-root cases. Several generalizations in which the notion of constacyclicity has been extended are also provided. Among others, we will present applications and open directions for future research.
Modules close to the automorphism-invariant and coinvariant
Truong Cong Quynh
Danang University of Education and VIASM
Abstract. In this talk, we study about the class of automorphism-invariant modules and the class of automorphism-coinvariant modules. Moreover, we generalized them to X-automorphism-invariant modules, X-idempotent-invariant modules... . We also give some new results of them with the relative structure rings.
A note on ring whose maximal right ideals are finitely generated
Le Van Thuyet
College of Education - Hue University and VIASM
We note that “maximal right ideal => cocritical right ideal => comonoform right ideal => completely prime right ideal”, but the converses are not true, in general. We also have that “maximal right ideal => completely prime right ideal => Michler-prime right ideal => Koh-prime right ideal”, but the converses are not true, in general. For a ring R, R is right Noetherian iff every Koh-prime (Michler-prime, completely prime, comonoform, cocritical) right ideal of R is finitely generated. It is very natural to discuss on the question: Is the ring R is right Noetherian iff every maximal right ideal of R is finitely generated. In this talk, we will discuss on some results of a ring whose maximal right ideals are finitely generated.
On essentially injective modules
Nguyen Thi Thu Ha
Industrial University of Ho Chi Minh City
A module M is called an essentially injective module if for every submodule A of N, any homomorphism f : A ! M with essential kernel can be extended to N. In this talk, we study on basic results of essentially injective modules. In particular, the invariant of essentially injective modules via injective envelope are obtained.
Characterizations of rings using quasi-injective modules
Phan Dan
Faculty of General Education, Hong Bang International
Let M be a right R-module. M is called an EES-module, if for every monomorphism f from M to N, there exists a direct summand E of M such that E is an essential extension f (M), where N in Mod-R. In this talk we provide a characterization of hereditary artinian serial rings in terms of EES-modules as follows:
Theorem. For any ring R, the following statements are equivalent:
- R is a hereditary artinian serial ring.
- R is a right nonsingular ring with essential right socle and every semisimple nonsingular right R-module is an ESS-module.
- R is a right nonsingular ring with essential right socle and every right nonsingular quasi-injective R-module is an EES-module
Finally, using the concept and above result, we get some decomposition theorems for right nonsingular rings over which every nonsingular quasi-injective right R-module is an EES-module.
Some characterizations of A-C3 modules
Tran Hoai Ngoc Nhan and Truong Thi Thuy Van
University of Technology Education, Vinh Long
Let A be a class of right modules over a ring R and closed under isomor- phisms. We call that a right R-module M is an A-C3 module if, whenever A in A and B in A are submodules of M with A and B direct summands of M and the intersection of A and B is zero then direct sum A + B is a direct summand of M. In this talk, some properties and characterizations of A-C3 modules are provided. For example, let M be a right R-module and A, a class of artinian right R-modules and closed under isomorphisms and summands. If every submodule of M is A-projective, then M is an A-C3 module if and only if M satisfies C2 for the class A, if and only if M have the summand sum property for the class A. Moreover, let A be a class of right R-modules and closed under isomorphisms and summands. Then all modules A in A are injective if and only if every right R-module is A-C3. This is joint work with Abyzov Adel Nailevich and Truong Cong Quynh.
A note on automorphism-invariant modules
Dao Thi Trang
Ho Chi Minh City University of Food and Industry
In this report, we will introduce some results on automorphism-invariant modules. We show that a ring R is quasi Frobenius if and only if R is a right automorphism-invariant, right ef-extending ring with maximum condition on right annihilators.
Overview on Code-based cryptography
Abhay Kumar Singh
IIT(ISM) Dhanbad
In a public-key cryptosystem (PKC) we consider two people who want to exchange a secret key, we call the constructor Bob and the second person Alice. Bob constructs a private key and public key, which he publishes. Alice who wants to send a message to Bob, uses the public key to encrypt her message and sends the cipher to Bob. Bob can decrypt the encrypted message with the private key. Eve, the eavesdropper, only sees the public key and the encrypted message. For the cryptosystem to be secure, it should be infeasible for Eve to reconstruct the message.
Code-based cryptography first came up in the 70's by the work of McEliece.The McEliece system in its original version uses a binary irreducible Goppa code,which is represented by a generator matrix G and can correct up to t errors. Instead of publishing the generator matrix directly, one publishes a scrambled matrix G0 = SGP, where S is an invertible matrix and P is a permutation matrix. In 1986, Niederreiter proposed a knapsack-type PKC based on error correcting codes. This proposal was later shown to have a security equivalent to McEliece's system. Niederreiter suggested to use generalized Reed-Solomon (GRS) codes, but the Niederreiter system (and therefore also the McEliece system) is broken when using GRS codes by the attack of Sidelnikov and Shestakov in 1992. The McEliece cryptosystem in its original version using Goppa codes is still unbroken, but has the main drawback of having large key sizes.Code-based system has been designed on many codes like Reed-Muller code, Algebraic code, LDPC, MDPC code, Reed Solomon code, polar code, QC-code, Convolution code. Many of these system has broken.
In 2010, Christian Wieschebrink proposed a new structural attack on the McEliece/Niederreiter public key cryp- tosystem based on subcodes of generalized Reed-Solomon codes proposed by Berger and Loidreau is described (2003, How to mask the structure of codes for cryptographic use). Baldi, Bianch, Chiaraluce, Rosenthal Schipani pro- posed "A variant of the McEliece cryptosystem with increased public key security" on Reed-Solomon code. Again in 2016 Baldi, Bianchi, Chiaraluce, Rosenthal and Schipani proposed a variant of the McEliece cryptosys-tem, in order to reconsider the use of GRS codes. Recently in 2019, there is another cryptosystem has designed by Rosenthial on Extended Reed-Solomon code.
DNA Cyclic Codes of Finite Rings
Abhay Kumar Singh
Department of Mathematics and Computing
Indian Institute of Technology (Indian School of Mines), Dhanbad
Deoxyribonucleic acid (DNA) contains genetic instructions for the structure and biological developments of life. It has the information on how the biological cell runs, reproduces builds and repairs itself. DNA strands sequences consist of four nucleotides; two purines: adenine (A) and guanine (G), and two pyrimidines: thymine (T) and cytosine (C).
The design of DNA strands has many applications in genetics, bioengineering and DNA computing. For example, application of biomolecular computing is the design of DNA chips for mutational analysis. DNA strands focuses on constructing large sets of DNA codewords with prescribed minimum Hamming distance. Furthermore, cyclic DNA computing has generated great interest because they have more storage capacity than silicon based computing systems, and this motivates many researchers to study it.
We have discussed the DNA code construction of general length over Z4+vZ4, v2=v. Further, GC-weight of DNA codes over is discussed and examples of reversible cyclic codes are provided. Later, we have obtained the structure of DNA cyclic codes of odd lengths over the rings R2=F2[u,v]/<u2-1,v3-v,uv-vu>, R3=Z4[u]/ <u2-1>, R4=F2+uF2+vF2+uvF2+v2F2+uv2F2, u2=0, v3=v. A necessary and sufficient condition is also determined for a cyclic code of arbitrary length over R=F2+uF2 (u2=1) to be a reversible cyclic code. Further, we have established a direct link between R2 (or R4) and 64 codons of amino acids of living organisms by introducing a Gray map from R2 (or R4) to R. We have also studied the GC-content of R3 and their deletion distance and also computed some examples of cyclic DNA codes with GC content and their respective deletion distance. The reversible and the reversible-complement codes over R4 are also investigated us. Finally, we have discussed the binary images of cyclic codes over R2 and R4 .