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An Algebraic Invariant for Free Convolutional Codes over Finite Local Rings

Mohammed El Oued

Abstract

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.

An Algebraic Invariant for Free Convolutional Codes over Finite Local Rings

Abstract

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.
Paper Structure (9 sections, 8 theorems, 14 equations)

This paper contains 9 sections, 8 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries results
  3. Convolutional codes over a finite ring
  4. Reduced integral degree matrices
  5. The Residual Structural Polynomial
  6. Definition and Invariance Theorem
  7. Characterization of Intrinsic Catastrophicity
  8. Duality and Catastrophicity Invariants
  9. conclusion

Key Result

Proposition 2.1

Let $G(D)\in{\mathbb Z}_{p^r}^{k\times n}[D]$. If an irreducible polynomial $P$ divides all $k\times k$ minors of $G$, then there exists a unimodular matrix $T(D)$ such that $P$ divides one row of $T(D)G(D)$.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Definition 3.1
  • ...and 11 more