[Paper Review] Error-Trellis Construction for Tailbiting Convolutional Codes

Masato Tajima, Koji Okino|arXiv (Cornell University)|Jan 21, 2012

Error Correcting Code Techniques10 references3 citations

TL;DR

This paper presents a novel error-trellis construction for tailbiting convolutional codes by leveraging the syndrome former's state behavior, ensuring the syndrome former starts and ends in the same state. It establishes a direct correspondence between code subtrellises and error subtrellises, derives a cyclic scalar parity-check matrix, and introduces backward error-trellises for time-reversed decoding, enabling low-complexity decoding with structured parity-check matrices.

ABSTRACT

In this paper, we present an error-trellis construction for tailbiting convolutional codes. A tailbiting error-trellis is characterized by the condition that the syndrome former starts and ends in the same state. We clarify the correspondence between code subtrellises in the tailbiting code-trellis and error subtrellises in the tailbiting error-trellis. Also, we present a construction of tailbiting backward error-trellises. Moreover, we obtain the scalar parity-check matrix for a tailbiting convolutional code. The proposed construction is based on the adjoint-obvious realization of a syndrome former and its behavior is fully used in the discussion.

Motivation & Objective

• To develop a systematic error-trellis construction for tailbiting convolutional codes that mirrors the structure of code-trellises.
• To clarify the correspondence between code subtrellises in the code-trellis and error subtrellises in the error-trellis.
• To construct backward error-trellises for time-reversed path representation in decoding.
• To derive the general structure of the scalar parity-check matrix for tailbiting convolutional codes.
• To establish a cyclic, structured parity-check matrix analogous to the scalar generator matrix for tailbiting codes.

Proposed method

• Uses the adjoint-obvious realization (observer canonical form) of the syndrome former $ H^T(D) $ to model error propagation.
• Defines the syndrome-former state $ \mathbf{\sigma}_k $ using memory element contents $ \sigma_{kp}^{(q)} $, ensuring state consistency across time.
• Imposes the condition $ \mathbf{\sigma}_0 = \mathbf{\sigma}_N $ to define the tailbiting error-trellis, analogous to the code-trellis constraint.
• Applies a superposition rule involving dual states (syndrome-former states) to relate code and error subtrellises.
• Derives the scalar parity-check matrix $ H_{\text{scalar}} $ by enforcing cyclic boundary conditions on syndrome equations.
• Constructs backward error-trellises by reversing the time order of error paths, enabling alternative decoding paths.

Experimental results

Research questions

• RQ1 How can an error-trellis be systematically constructed for tailbiting convolutional codes such that the syndrome former starts and ends in the same state?
• RQ2 What is the precise correspondence between code subtrellises in the code-trellis and error subtrellises in the error-trellis?
• RQ3 How can backward error-trellises be constructed to represent error paths in time-reversed order?
• RQ4 What is the general structure of the scalar parity-check matrix for a tailbiting convolutional code?
• RQ5 How does the use of the adjoint-obvious syndrome former realization enable the derivation of structured parity-check matrices?

Key findings

• The tailbiting error-trellis is fully characterized by the condition $ \mathbf{\sigma}_0 = \mathbf{\sigma}_N $, ensuring consistent syndrome formation across the frame.
• A one-to-one correspondence exists between code subtrellises (with fixed initial/final encoder states) and error subtrellises (with fixed initial/final syndrome-former states).
• The scalar parity-check matrix $ H_{\text{scalar}} $ for a tailbiting convolutional code has a cyclic structure, analogous to the scalar generator matrix.
• The derived $ H_{\text{scalar}} $ is given by a circulant-like block matrix with blocks $ H_0, H_1, \dots, H_M $, arranged in a cyclic pattern across rows.
• The backward error-trellis construction enables representation of error paths in reverse time order, supporting alternative decoding strategies.
• The proposed method generalizes the scalar parity-check matrix construction to tailbiting codes using syndrome former state consistency and cyclic boundary conditions.