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Construction of observable and MDP convolutional codes with good decodable properties for erasure channels by I/S/O representations

Noemí DeCastro-García, Miguel V. Carriegos, Ángel Luis Muñoz Castañeda

TL;DR

The work tackles constructing observable convolutional codes with strong erasure-channel decodability using input/state/output (I/S/O) representations and group actions. It develops an algebraic framework where reachable, observable, and GDP (and conjecturally MDP) properties are preserved under three types of transformations on the I/S/O data, enabling generation of diverse codes from a single seed representation. Key contributions include formal invariance results for reachability, observability, and GDP under these actions, as well as concrete examples over small finite rings to illustrate non-equivalence and preserved decoding performance. The findings offer a principled method to design robust erasure-decoding convolutional codes with potential extensions to other algebraic settings and rings.

Abstract

This paper addresses the construction of observable convolutional codes that exhibit good performance with the available decoding algorithms for erasure channels. Our construction is based on the use of input/state/output (I/S/O) representations and the invariance of certain properties of linear systems under various group actions.

Construction of observable and MDP convolutional codes with good decodable properties for erasure channels by I/S/O representations

TL;DR

The work tackles constructing observable convolutional codes with strong erasure-channel decodability using input/state/output (I/S/O) representations and group actions. It develops an algebraic framework where reachable, observable, and GDP (and conjecturally MDP) properties are preserved under three types of transformations on the I/S/O data, enabling generation of diverse codes from a single seed representation. Key contributions include formal invariance results for reachability, observability, and GDP under these actions, as well as concrete examples over small finite rings to illustrate non-equivalence and preserved decoding performance. The findings offer a principled method to design robust erasure-decoding convolutional codes with potential extensions to other algebraic settings and rings.

Abstract

This paper addresses the construction of observable convolutional codes that exhibit good performance with the available decoding algorithms for erasure channels. Our construction is based on the use of input/state/output (I/S/O) representations and the invariance of certain properties of linear systems under various group actions.

Paper Structure

This paper contains 8 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Convolutional Codes
  4. Convolutional Codes and Linear Systems
  5. The Decoding Process Using I/S/O Representations
  6. Construction of observable and GDP convolutional codes by I/S/O representations
  7. Some Conjectures on MDP Construction of Convolutional Codes
  8. Conclusions

Key Result

Theorem 2.7

If for all such subspaces $U_{j_1, \ldots, j_\alpha}$, the condition $U_{j_1, \ldots, j_\alpha} \oplus \langle \Omega_{L+1}(A,C) \rangle = \mathbb{F}^{(L+1)(n-k)}$ holds, then it is possible to recalculate the state of the system (equation isoerasure) when we observe a window of length $(L+1)n$, pro

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2: cf. Definition 2.25, Mc
  • Definition 2.3: cf. Section 5.2, York, Remark 4.1, ros
  • Definition 2.4: cf. Theorem 5.1.1, York, Theorem 3.1, ros
  • Remark 1
  • Definition 2.5
  • Remark 2: cf. Lemma 5.3.5, York
  • Definition 2.6
  • Theorem 2.7: cf. Theorem 4.3., tesisvirtudes
  • Definition 2.8
  • ...and 17 more