Necessary and Sufficient Conditions for Minimal Linear Codes over the Ring $\mathbb{Z}_{p^l}$

Concepts de base

This paper establishes necessary and sufficient conditions for a k-dimensional linear code over the ring $\mathbb{Z}_{p^l}$ to achieve minimality. It also provides upper and lower bounds for the existence of minimal linear codes of a given dimension.

Résumé

The paper explores the structure of modules over the ring $\mathbb{Z}_{p^l}$ and uses this understanding to derive necessary and sufficient conditions for the minimality of linear codes over this ring.

Key highlights:

• Defines the concepts of support, cover, Hamming distance, weight, and minimal codewords for linear codes over $\mathbb{Z}_{p^l}$.
• Proves that the orthogonal complement of a root word (a vector with at least one unit component) is a free module of dimension k-1, while the orthogonal complement of a non-root word is a finitely generated module.
• Establishes that the double orthogonal of a vector is the submodule generated by that vector.
• Provides a necessary and sufficient condition for a codeword to be minimal: the submodule generated by the vectors orthogonal to the codeword must equal the full orthogonal complement.
• Extends the concept of the parameter n(k;q) from finite fields to the ring $\mathbb{Z}_{p^l}$, proving that minimal linear codes of dimension k exist if and only if the length n is at least n(k;$p^l$).
• Derives upper and lower bounds for n(k;$p^l$), showing that $(k-1)p^l + p^{l-k} < n(k;p^l) \leq \frac{k(k-1)}{2}(p^l + p^{l-1} - 2) + k$ for $k \geq 3$, and $p^l + p^{l-2} + 1 < n(k;p^l) \leq \frac{k(k-1)}{2}(p^l + p^{l-1} - 2) + k$ for $k = 2$.

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Idées clés tirées de

Generalization the parameters of minimal linear codes over the ring $\mathbb{Z}_{p^l}$

by Biplab Chatt... à arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09561.pdf

Questions plus approfondies

How can the results in this paper be extended to linear codes over other finite commutative rings beyond $\mathbb{Z}_{p^l}$

The results presented in this paper can be extended to linear codes over other finite commutative rings beyond $\mathbb{Z}_{p^l}$ by adapting the fundamental principles and techniques established for minimal linear codes. Since the structure of modules over commutative rings shares similarities with vector spaces over fields, the concepts of minimal codewords, generator matrices, and Hamming distances can be generalized to codes over different finite commutative rings. By considering the specific properties and algebraic structures of the new rings, similar necessary and sufficient conditions for achieving minimality can be derived. The key lies in understanding the underlying algebraic properties of the ring and how they influence the construction and characterization of minimal linear codes.

What are the implications of these minimal linear codes in the context of secret sharing schemes and secure multi-party computation

Minimal linear codes play a crucial role in secret sharing schemes and secure multi-party computation due to their ability to efficiently encode information while ensuring a high level of security. By utilizing minimal linear codes, it becomes possible to distribute secret information among multiple parties in such a way that only authorized subsets of participants can reconstruct the original data. The minimality property ensures that the codes have optimal error-correcting capabilities while minimizing redundancy, making them ideal for applications where data privacy and integrity are paramount. Additionally, the construction and analysis of minimal linear codes enable the development of efficient decoding algorithms and access structures for secure communication and computation protocols.

Can the techniques developed in this work be applied to study the properties of other types of codes, such as non-linear codes, over finite commutative rings

The techniques developed in this work for studying minimal linear codes over finite commutative rings like $\mathbb{Z}_{p^l}$ can be extended to investigate the properties of other types of codes, including non-linear codes, over similar algebraic structures. While the focus of this paper is on linear codes and their minimality, the underlying principles of module theory and algebraic coding theory can be applied to analyze the characteristics of non-linear codes over finite commutative rings. By adapting the concepts of support, cover, and Hamming distance to non-linear codes, researchers can explore the minimality and error-correcting capabilities of these codes in the context of different algebraic structures. This extension opens up avenues for studying a broader range of codes and their applications in various cryptographic and communication systems.