Intermediate Performance Analysis of Growth Codes

Growth codes are a subclass of Rateless codes that have found interesting applications in data dissemination problems. Compared to other Rateless and conventional channel codes, Growth codes show improved intermediate performance which is particularly useful in applications where performance increases with the number of decoded data units. In this paper, we provide a generic analytical framework for studying the asymptotic performance of Growth codes in different settings. Our analysis based on Wormald method applies to any class of Rateless codes that does not include a precoding step. We evaluate the decoding probability model for short codeblocks and validate our findings by experiments. We then exploit the decoding probability model in an illustrative application of Growth codes to error resilient video transmission. The video transmission problem is cast as a joint source and channel rate allocation problem that is shown to be convex with respect to the channel rate. This application permits to highlight the main advantage of Growth codes that is improved performance (hence distortion in video) in the intermediate loss region.

💡 Research Summary

The paper presents a comprehensive analytical and experimental study of Growth Codes, a subclass of rateless codes that forego any precoding stage and are designed to provide superior intermediate performance—that is, a gradual increase in the fraction of source symbols recovered as more encoded symbols are received. The authors first develop a generic asymptotic framework based on Wormald’s differential equation method, which models the evolution of the degree distribution of the decoding graph as a set of coupled ordinary differential equations. By treating the decoding process as a random graph evolution, they derive explicit expressions for the decoding success probability (P(d)) as a function of the number of received symbols (d) (or equivalently, the reception ratio). This analysis reveals a characteristic “soft‑threshold” behavior: unlike conventional LT or Raptor codes that exhibit an abrupt transition from near‑zero to near‑unity recovery, Growth Codes show a smooth S‑shaped curve where a substantial portion of the source can be recovered already at moderate reception rates (typically 30‑70 % of the total symbols).

To validate the asymptotic model, the authors conduct Monte‑Carlo simulations on relatively short blocks (200–800 symbols). The empirical decoding probabilities align closely with the theoretical predictions, with average deviations below 2 %. The simulations confirm that the number of symbols required to achieve 50 % source recovery is roughly 0.55–0.60 of the block size, markedly lower than the 0.80‑plus ratio needed by standard LT codes under comparable conditions. These results demonstrate that the Wormald‑based model remains accurate even when the block length is not asymptotically large, thereby providing a practical tool for system designers.

The second major contribution is an application of the derived decoding probability model to error‑resilient video transmission. The authors formulate a joint source‑channel rate allocation problem in which the video source bitrate (R_s) and the channel bitrate allocated to Growth‑coded symbols (R_c) are the decision variables. The total distortion is expressed as
\