### Self-dual and LCD double circulant and double negacirculant codes over a family of finite rings Fq[v1, v2, … ,vt] (part 1 and 2)

Thời gian: 14:00 đến 15:30 ngày 18/07/2022, 14:00 đến 15:30 ngày 22/07/2022,

Địa điểm: C101, VIASM

Báo cáo viên: Bhanu Pratap Yadav

Tóm tắt:

we discuss self-dual double circulant codes, LCD double circulant codes and double negacirculant codes over a family of finite rings \$R_{t} = \mathbb F_{q} + v_{1} \mathbb F_{q} + v_{2}\mathbb F_{q} + \ldots + v_{t}\mathbb F_{q}\$; \$(v_{i}^{2} = v_{i}, v_{i}v_{j} = v_{j}v_{i}=0\$, \$i,j= 1,2,\ldots,t\$, \$i \neq j),\$ where \$q\$ is an odd prime power. We obtain necessary and sufficient conditions for a double circulant code (double negacirculant code) to be a self-dual code and a double circulant code  (double negacirculant code) to be an LCD code. We derive a formula to determine the total number of self-dual double circulant codes, LCD double circulant codes, self-dual double negacirculant codes and LCD double negacirculant codes over the ring \$R_{t}\$. We  also find the distance bounds for self-dual double circulant codes and LCD double circulant codes over \$R_{t}\$. Moreover, we construct a Gray map, and prove that the families of self-dual double circulant codes and LCD double circulant codes under this Gray map are asymptotically good. Finally, we give some examples for small length codes under the Gray image of LCD double circulant codes and self-dual circulant codes over \$\mathbb{F}_{5} + u \mathbb{F}_{5}\$.