Certain binary minimal codes constructed using simplicial complexes

TL;DR: Sure, here is the TLDR: The manuscript studies binary minimal codes constructed using simplicial complexes over the non-chain ring $ \mathcal{R} = \frac{\mathbb{F}_2[u]}{\langle u^3 - u\rangle} $. Codes are constructed using the Delta sets and the ordered finite multiset $ D $. Conditions for minimality and self-orthogonality are obtained.

Abstract: In this manuscript, we work over the non-chain ring $ \mathcal{R} = \frac{\mathbb{F}_2[u]}{\langle u^3 - u\rangle} $. Let $ m\in \mathbb{N} $ and let $ L, M, N \subseteq [m]: = \{1, 2, \dots, m\} $. For $ X\subseteq [m] $, define $ \Delta_X: = \{v \in \mathbb{F}_2^m : \text{Supp}(v)\subseteq X\} $ and $ D: = (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3 $, an ordered finite multiset consisting of elements from $ \mathcal{R}^m $, where $ D_1\in \{\Delta_L, \Delta_L^c\}, D_2\in \{\Delta_M, \Delta_M^c\}, D_3\in \{\Delta_N, \Delta_N^c\} $. The linear code $ C_D $ over $ \mathcal{R} $ defined by $ \{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \} $ is studied for each $ D $. Further, we also consider simplicial complexes with two maximal elements. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $ \mathbb{F}_{2} $-functional of $ \mathcal{R} $. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.