Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.13468 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915798241509376 |
|---|---|
| author | Oued, Mohammed El |
| author_facet | Oued, Mohammed El |
| contents | This paper investigates the algebraic structure of free convolutional codes over the finite local ring Z_{p^r}. We introduce a new structural invariant, the Residual Structural Polynomial, denoted by Delta_p(C) in F_p[D]. We construct this invariant via encoders which are reduced internal degree matrices (RIDM). We formally demonstrate that Delta_p(C) is an intrinsic characteristic of the code, invariant under equivalent RIDMs. A central result of this work is the establishment that Delta_p(C) serves as an algebraic criterion for intrinsic catastrophicity: we prove that a free code C admits a non-catastrophic realization if and only if Delta_p(C) is a monomial of the form D^s. Furthermore, we establish a fundamental duality theorem, proving that Delta_p(C) = Delta_p(C^perp). This result reveals a deep structural symmetry, showing that the "catastrophicity" of a free code is preserved under orthogonality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13468 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Algebraic Invariant for Free Convolutional Codes over Finite Local Rings Oued, Mohammed El Information Theory 94B10, 11T71, 13M05 E.4; G.2.1 This paper investigates the algebraic structure of free convolutional codes over the finite local ring Z_{p^r}. We introduce a new structural invariant, the Residual Structural Polynomial, denoted by Delta_p(C) in F_p[D]. We construct this invariant via encoders which are reduced internal degree matrices (RIDM). We formally demonstrate that Delta_p(C) is an intrinsic characteristic of the code, invariant under equivalent RIDMs. A central result of this work is the establishment that Delta_p(C) serves as an algebraic criterion for intrinsic catastrophicity: we prove that a free code C admits a non-catastrophic realization if and only if Delta_p(C) is a monomial of the form D^s. Furthermore, we establish a fundamental duality theorem, proving that Delta_p(C) = Delta_p(C^perp). This result reveals a deep structural symmetry, showing that the "catastrophicity" of a free code is preserved under orthogonality. |
| title | An Algebraic Invariant for Free Convolutional Codes over Finite Local Rings |
| topic | Information Theory 94B10, 11T71, 13M05 E.4; G.2.1 |
| url | https://arxiv.org/abs/2602.13468 |