[Paper Review] Explicit and Efficient Constructions of Coding Schemes for the Binary Deletion Channel and the Poisson Repeat Channel

Roni Con, Amir Shpilka|arXiv (Cornell University)|Sep 23, 2019

Advanced biosensing and bioanalysis techniques18 references4 citations

TL;DR

This paper presents the first explicit construction of error-correcting codes for the binary deletion channel (BDC) and the Poisson repeat channel (PRC), achieving a rate of $(1-p)/16$ for BDC and $rac{ u}{17}$ for PRC with parameter $ u$, using polynomial-time encoding and decoding. The construction leverages a concatenated coding scheme with an inner code robust to synchronization errors, enabling high-probability decoding under random deletions or repetitions.

ABSTRACT

This work gives an explicit construction of a family of error correcting codes for the binary deletion channel and for the Poisson repeat channel. In the binary deletion channel with parameter $p$ (BDC$_p$) every bit is deleted independently with probability $p$. A lower bound of $(1-p)/9$ is known on the capacity of the BDC$_p$ \cite{mitzenmacher2006simple}, yet no explicit construction is known to achieve this rate. We give an explicit family of codes of rate $(1-p)/16$, for every $p$. This improves upon the work of Guruswami and Li \cite{guruswami2017efficiently} that gave a construction of rate $(1-p)/120$. The codes in our family have polynomial time encoding and decoding algorithms. Another channel considered in this work is the Poisson repeat channel with parameter $λ$ (PRC$_λ$) in which every bit is replaced with a discrete Poisson number of copies of that bit, where the number of copies has mean $λ$. We show that our construction works for this channel as well. As far as we know, this is the first explicit construction of an error correcting code for PRC$_λ$.

Motivation & Objective

• To close the gap between known capacity lower bounds and explicit code constructions for the binary deletion channel (BDC).
• To provide the first explicit construction of error-correcting codes for the Poisson repeat channel (PRC), a channel closely related to the BDC but previously lacking explicit code designs.
• To achieve a higher rate than prior explicit constructions for the BDC, improving upon the $(1-p)/120$ rate of Guruswami and Li.
• To design a code with polynomial-time encoding and decoding algorithms that maintain high reliability under random deletions or Poisson-distributed repetitions.

Proposed method

• A concatenated coding scheme is employed, combining an outer code with an inner code designed to be robust against synchronization errors.
• The inner code is constructed using a structure that allows reliable decoding even when bits are deleted or repeated according to a Poisson distribution.
• The decoding algorithm uses buffer-based recovery to detect and correct errors caused by deletions or repetitions, with probabilistic analysis of error events.
• The analysis bounds the probability of decoding failure using Poisson tail inequalities and concentration bounds, ensuring exponentially small error rates.
• The construction is adapted to the PRC by modeling bit repetitions as Poisson-distributed copies and adjusting the decoding thresholds accordingly.
• Rate analysis is performed using the inner code rate and the expected number of surviving symbols, leading to a final code rate of $\mathcal{R} > \frac{\lambda}{17}$ for the PRC.

Experimental results

Research questions

• RQ1 Can an explicit, high-rate error-correcting code be constructed for the binary deletion channel that achieves a rate approaching the known capacity lower bound of $(1-p)/9$?
• RQ2 Is it possible to design the first explicit error-correcting code for the Poisson repeat channel, which models random bit repetition?
• RQ3 What is the maximum achievable rate for explicit codes in the BDC and PRC under polynomial-time encoding and decoding constraints?
• RQ4 How can a code be designed to be robust against both deletions and insertions (or repetitions) in a probabilistic setting with high-probability decoding?

Key findings

• The paper constructs explicit codes for the BDC with rate $(1-p)/16$, improving upon the prior explicit construction of rate $(1-p)/120$ by Guruswami and Li.
• For the Poisson repeat channel with parameter $\lambda$, the construction achieves a rate exceeding $\lambda/17$, marking the first explicit code for this channel.
• The decoding algorithm runs in polynomial time and succeeds with probability $1 - \exp(-\Omega(m))$, where $m$ is the block length of the outer code.
• The inner code is shown to be robust against both deletions and spurious insertions, with error probability bounded using Poisson tail inequalities.
• The construction is robust under the edit distance adversary, simplifying the analysis and strengthening the decoding guarantees.
• The rate analysis confirms that the overall code rate approaches $\lambda/17$ as the block length increases, with explicit parameters chosen to ensure high reliability.