Convolutional-Code-Specific CRC Code Design

TL;DR

This paper introduces an algorithm to design CRC codes optimized for specific convolutional codes, significantly reducing undetected errors by leveraging error pattern enumeration and an equivalent catastrophic code approach.

Contribution

It presents a novel algorithm that optimizes CRC codes for specific convolutional codes, improving detection performance and reducing code length.

Findings

Optimized CRC codes reduce undetected error probability.

Significant reduction in CRC length needed for target error detection.

Algorithm applicable to commonly used convolutional codes.

Abstract

Cyclic redundancy check (CRC) codes check if a codeword is correctly received. This paper presents an algorithm to design CRC codes that are optimized for the code-specific error behavior of a specified feedforward convolutional code. The algorithm utilizes two distinct approaches to computing undetected error probability of a CRC code used with a specific convolutional code. The first approach enumerates the error patterns of the convolutional code and tests if each of them is detectable. The second approach reduces complexity significantly by exploiting the equivalence of the undetected error probability to the frame error rate of an equivalent catastrophic convolutional code. The error events of the equivalent convolutional code are exactly the undetectable errors for the original concatenation of CRC and convolutional codes. This simplifies the computation because error patterns do not need to be individually checked for detectability. As an example, we optimize CRC codes for a commonly used 64-state convolutional code for information length k=1024 demonstrating significant reduction in undetected error probability compared to the existing CRC codes with the same degrees. For a fixed target undetected error probability, the optimized CRC codes typically require 2 fewer bits.