Research Projects

In the last 30 years, many researchers contributed a lot in error-correcting codes in the sense of giving the complete algebraic structures of different types of linear codes over various finite commutative Frobenius rings along with computing their Hamming distance in most of the cases but a very few on computing the Lee distance. One of the objectives is obtain complete Lee distance distributions of certain types of linear codes over some finite commutative local rings and tried to see if any important class of linear codes (in the Hamming metric setting) over finite fields can be obtained through Gray maps. As of now, the study of computing the Lee distance of different types of linear codes over different finite commutative chain rings mostly engage with the rings of characteristics 2^k. In fact, it will be much more difficult and hence a very challenging problem to move on to the rings with characteristics p^k for odd prime p with k a positive integer. Among various metrics defined on rings, the Lee metric is not only more significant in the sense of finding good codes over finite fields through Gray maps also it has been proposed for a wide range of applications such as multidimensional burst error correction, VLSI implementation, DNA based storage etc. Even some recent studies of information set decoding (ISD) algorithms in the Lee metric validates its use in the code-based cryptography as a promising candidate of post quantum cryptosystems. So, finding good classes of Lee metric codes with a fast decoding algorithm will also be an important step. Also, it would be interesting to study whether we can use the Lee metric in other areas like codes for Symbol-pair read channels etc. The mathematical theory of quantum error-correcting codes has aided in the construction of a large family of quantum codes. A prominent example of quantum codes are the stabilizer codes. The theory of stabilizer codes has been developed via considering binary codes over symplectic inner product and via the Hermitian self- orthogonal codes over GF(4). The method of obtaining stabilizer codes via Hermitian self-orthogonal codes over GF(4) is a well-studied construction. Another objective is to explore the possibility to develop the theory of quantum error correcting codes in the Lee metric setting and obtain stabilizer codes via Lee self-orthogonal codes over GF(4).