Varshamov-Tenengolts Codes Research Articles

A New Version of q-Ary Varshamov-Tenengolts Codes With More Efficient Encoders: The Differential VT Codes and The Differential Shifted VT Codes

A New Version of q-Ary Varshamov-Tenengolts Codes With More Efficient Encoders: The Differential VT Codes and The Differential Shifted VT Codes

On the Positional Single Error Correction and Double Error Detection in Racetrack Memories

In the era of non-volatile memories, the racetrack memory is a promising technology to pack hundreds of bits in a magnetic nanowire. A solid-state read head is grown alongside the nanowire to sense individual bits which are pushed across the head by a shifting force. However, the probabilistic nature of this shifting movement inflicts positional errors. Therefore, robust and low-cost error correcting codes are essential for a reliable alternative in on-chip memories and storage applications. Recent works focus on Varshamov-Tenengolts (VT) codes which can correct all single bit insertions and deletions. However, VT codes are incapable of detecting multiple deletions/insertions. Because a positional error corrupts multiple data words, multi-bit positional error detection is critical for racetrack memories. In this article, we propose a novel positional single error correction and double error detection (P-SECDED) code in the context of racetrack memories with a single read head. In particular, we adopt a postamble-based approach where a VT-encoded codeword appends a carefully selected bit-pattern, stored on the racetrack. We rigorously analyze the limitations of the postamble method, and deduce a criterion for postamble selection. We further provide a methodology to optimize this postamble selection in order to correct all single-bit errors and as much of two-bit errors as possible. Finally, we prove that all incurable two-bit errors are successfully detected. To the best of our knowledge, this work is the first attempt to provide P-SECDED fault-tolerance to three-dimensional racetrack memories which cannot afford multiple read heads.

On Prefixed Varshamov-Tenengolts Codes for Segmented Edit Channels

The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels.

Multilayer Codes for Synchronization From Deletions and Insertions

Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder's sequence X differs from the decoder's sequence Y by a small number of edits (deletions and insertions). The goal is to construct a message M, to be sent via a one-way error free link, such that the decoder can reconstruct X using M and Y. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expected list size of the decoder under certain assumptions, and validate its performance via numerical simulations. We also consider an alternative decoder that uses only the constraints from the VT codes (i.e., does not require a linear code), and has a smaller redundancy at the expense of a slightly larger average list size.

Nonasymptotic Upper Bounds on Binary Single Deletion Codes via Mixed Integer Linear Programming

The size of the largest binary single deletion code has been unknown for more than 50 years. It is known that Varshamov–Tenengolts (VT) code is an optimum single deletion code for block length ; however, only a few upper bounds of the size of single deletion code are proposed for larger n. We provide improved upper bounds using Mixed Integer Linear Programming (MILP) relaxation technique. Especially, we show the size of single deletion code is smaller than or equal to 173 when the block length n is 11. In the second half of the paper, we propose a conjecture that is equivalent to the long-lasting conjecture that “VT code is optimum for all n”. This equivalent formulation of the conjecture contains small sub-problems that can be numerically verified. We provide numerical results that support the conjecture.

An Efficient Syndrome Calculator for Varshamov–Tenengolts Codes in Vertical Racetrack Memory

Spintronic racetrack memory stores binary information in magnetic domains, isolated with domain walls and movable within a nanowire. Because the domain wall motion is probabilistic, Varshamov–Tenengolts (VT) codes have been applied to such devices due to their ability to correct single-bit deletion and insertion. The complexity of the VT encoder and decoder is largely because of a sequential syndrome calculator that employs modulo-additions. Previous implementations avoided this complexity by restricting the racetrack size to a specific number of magnetic domains. In this letter, we describe an efficient implementation of the syndrome calculator with a novel steering algorithm, which not only alleviates the above-mentioned restrictions but also reduces adder size and the required number of operations, exploiting modulo-arithmetic symmetries in VT codes. We synthesize the syndrome calculator with varying number of domains to show that the algorithm is scalable, increases areal efficiency, slashes processing time, and lowers the power consumption of a VT encoder and decoder.

List Decoding of Insertions and Deletions

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions.

Unweighted linear congruences with distinct coordinates and the Varshamov–Tenengolts codes

In this paper, we first give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov--Tenengolts code $VT_b(n)$ with Hamming weight $k$, that is, with exactly $k$ $1$'s. The Varshamov--Tenengolts codes are an important class of codes that are capable of correcting asymmetric errors on a $Z$-channel. As another application, we derive Ginzburg's formula for the number of codewords in $VT_b(n)$, that is, $|VT_b(n)|$. We even go further and discuss connections to several other combinatorial problems, some of which have appeared in seemingly unrelated contexts. This provides a general framework and gives new insight into all these problems which might lead to further work.

Exact Reconstruction From Insertions in Synchronization Codes

This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion/insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion/deletion channels and probabilistic channels.

Nonasymptotic Upper Bounds for Deletion Correcting Codes

Explicit nonasymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for q-ary alphabet and string length is shown to be of size at most (qn-q)/{(q-1)(n-1)}. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The nonasymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known nonasymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.