Reassembly Codes for the Chop-and-Shuffle Channel

Abstract

We study the problem of retrieving data from a channel that breaks the input sequence into a set of unordered fragments of random lengths, which we refer to as the chop-and-shuffle channel. The length of each fragment follows a geometric distribution. We propose nested Varshamov-Tenengolts (VT) codes to recover the data. We evaluate the error rate and the complexity of our scheme numerically. Our results show that the decoding error decreases as the input length increases, and our method has a significantly lower complexity than the baseline brute-force approach. We also propose a new construction for VT codes, quantify the maximum number of the required parity bits, and show that our approach requires fewer parity bits compared to known results.

Adversarial Torn-paper Codes

“…Multiple channel models have recently been suggested and studied based on this property. An assumption of overlap in read substrings and (near) uniform coverage leads to the problem of string reconstruction from substring composition [3], [4], [7], [13], [21], [22], [27], [29]; on the contrary, assuming no overlap in read substrings leads to the torn-paper problem [23], [25], [30], a problem closely related to the shuffling channel [16], [17], [28], [32]. This problem is motivated by DNA-based storage systems, where the information is stored in synthesized strands of DNA molecules.…”

“…In the torn-paper channel [25], [30], also known as the chop-and-shuffle channel [23], a long information sequence is The first, second, and fourth authors contributed equally to this work.…”

“…Given the lengths' distribution, the goal is to determine the channel capacity and devise efficient coding techniques. The geometric distribution was first studied in [30], and later in [23] using the Varshamov-Tenengolts (VT) codes [31]. The work [25] considered almost arbitrary distributions while, additionally, extending the problem by introducing incomplete coverage, i.e., assuming some of the substrings are deleted with some probability.…”

Reconstruction from Substrings with Partial Overlap

“…Notably, when observations are composed of consecutive substrings, the reconstruction from substring-compositions problem [1], [1], [4], [10], [12], [17], [19], [20], [25], [27], [31], [32] and the torn-paper problem [2], [21], [22], [28] (a problem closely related to the shuffling channel [11], [13], [26], [30]) have received significant interest in the past decade due to applications in DNA-or polymer-based storage systems, resulting from contemporary sequencing technologies [4], [9], [20]. The former arises from an idealized assumption of full overlap (and uniform coverage) in read substrings, while the latter results from an assumption of no overlap; in applications, this models the question of whether the complete information string may be replicated and uniformly segmented for sequencing, or if segmentation occurs adversarially in the medium prior to sequencing.…”

Multi-strand Reconstruction from Substrings

“…Notably, when observations are comprised of unordered consecutive substrings, two distinct models have received significant interest in the past decade due to applications in DNA-or polymer-based storage systems, resulting from contemporary sequencing technologies [4], [12], [26]. The first is the reconstruction from substring-compositions problem [1], [4], [10], [14], [17], [23], [25], [26], [32], [34], [38], [39] (including extensions for erroneous observations [5], [12], [23], [39]), which arises from an idealized assumption of full overlap (and uniform coverage) in read substrings; the second is the torn-paper problem [2], [27], [28], [35] (a problem closely related to the shuffling channel [15], [18], [33], [37]), which results from an assumption of no overlap. In applications, the distinction models the question of whether the complete information string may be replicated and uniformly segmented for sequencing, or if segmentation occurs adversarially in the medium prior to sequencing.…”