Minimal linear codes have garnered significant attention in cryptography due to their essential role in secretsharing schemes, multiparty computation (MPC), and secure communication. While earlier studies primarily focused on minimal codes over finite fields, extending these codes to rings offers enhanced security, flexibility, and efficiency for various cryptographic applications. This article deals with minimal linear codes over finite commutative rings. Specifically, we handle a key question in minimal linear codes, which is determining the existence of an [m,k] minimal linear code withk≤m. In 2021, Lu,Wu and Cao proved that for some integerm(k;q), a minimal linear code of lengthmand dimensionkalways exists whenm≥m(k;q), providing upper and lower bounds form(k;q). This paper expands on these results by establishing both upper and lower bounds form(k;pl) andm(k;p1p2) over Zpland Zp1p2, respectively. We also present a necessary and sufficient condition for a linear code to achieve minimality when analyzed over the ring Zn. The fundamental question regarding minimal linear codes is whether a [m,k] minimal linear code exists, withkbeing less than or equal tom. Lu et al. demonstrated that there is a positive integerm(k;q) such that ifm≥m(k;q), a minimal linear code of lengthmand dimensionkover the finite field Fqmust exist (whereqis a prime power). They provided both upper and lower bounds form(k;q). In our study, we investigate the existence of minimal codes within an extended novel framework. We analyze the case of one-dimensional minimal codes over Znin depth and present improved upper bounds onm(k;q) specifically for the case whenk= 1.

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings$\mathbb {Z}_{p^{n}}$(where$p$is a prime) and$\mathbb {Z}_{p^{m}q^{n}}$(where$p < q$are distinct primes and$m\leq n$). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future.