Pré-Publication, Document De Travail Année : 2025

Linearized Polynomial Chinese remainder codes

Codes restes chinois sur les polynômes linéarisés

Résumé

der theorem tailored to these rings. This enables us to construct a wide range of linearized CRT codes with diverse parameters.

2. We present explicit constructions and examples of linearized CRT codes, highlighting their relation to known codes in the literature.

3. We propose a decoding algorithm for a special case of linearized CRT codes and extend it to a broader range of applications. This algorithm has an associated failure rate, which we analyze and discuss.

4. Finally, we study the parameters of these new codes to provide insights into their decoding properties and potential applications in various fields.

Organization: The remainder of this paper is structured as follows. Section 2 introduces rank and sum-rank metric codes and their applications. In Section 3, we discuss linearized polynomial rings over finite fields and the effective Chinese remainder theorem for these rings. Section 4 presents the construction of linearized CRT codes, along with examples and connections to existing codes. A decoding algorithm is proposed in Section 5, followed by an analysis of its failure rate and parameters. We extend this algorithm to a wider class in Section 6 together with the parameters and failure rate in this case.

Rank and sum-rank metric codes

In this section, we introduce the notation and foundational concepts for rank and sum-rank metric codes. For a more extensive presentation of rank metric codes, we refer to [2].

Let q be a power of a prime and let m ∈ N, we denote F q the field with q elements and F q m an extension of F q of degree m. The field F q m can be considered as a F q -vector space of dimension m, i.e.

) is a linear isomorphism in a way that every subspace W of V can be identified to a subspace M(W ) of matrices.

Conversely, every F q -linear subspace in M m×n (F q ) can be identified to a F qsubvector space of V .

Definition 1 Rank Metric: Let V and W be two finite dimensional F q -vector spaces and let c ∈ End(V, W ), we define the rank weight w r (c) of c as its rank

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hal-05062636 , version 1 (10-05-2025)

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  • HAL Id : hal-05062636 , version 1

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Philippe Gaborit, Camille Garnier, Olivier Ruatta. Linearized Polynomial Chinese remainder codes. 2025. ⟨hal-05062636⟩
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