arXiv subjects
Publications and source records attributed to Wajid M. Shaikh.
In this article, we present two new approaches to construct minimal linear codes of dimension $n+1$ over $\mathbb{F}_{3}$ using characteristic and ternary functions. We also obtain the weight distributions of these constructed minimal linear codes. We further show that a specific class of these codes violates Ashikhmin-Barg condition.
In this paper, we will give the generic construction of a binary linear code of dimension $n+3$ and derive the necessary and sufficient conditions for the constructed code to be minimal. Using generic construction, a new family of minimal binary linear code will be constructed from a special class of Boolean functions violating the Ashikhmin-Barg condition. We also obtain the weight distribution of the constructed minimal binary linear code.
In this paper, we develop the python code for generating unit graph $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$, for any integers $m\ \& \ n$. For any prime $r$, we construct $r$-ary linear codes from the incidence matrix of the unit graph $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$, where $n \ \& \ m$ are either power of prime or product of power of primes. We also prove the minimum distance of dual of the constructed codes as either 3 or 4. Finally, we state conjectures two on linear codes constructed from the unit graph $G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})$, for any integer $m\ \& \ n$.
In this paper, we consider the unit graph $G(\mathbb{Z}_{n})$, where $n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ and $p_{1}, p_{2}, p_{3}$ are distinct primes. For any prime $q$, we construct $q$-ary linear codes from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ with their parameters. We also prove that the dual of the constructed codes have minimum distance either 3 or 4. Lastly, we stated two conjectures on diameter of unit graph $G(\mathbb{Z}_{n})$ and linear codes constructed from the incidence matrix of the unit graph $G(\mathbb{Z}_{n})$ for any integer $n$.