Efficient Systematic Encoding of Non-binary VT Codes

Abstract

Varshamov-Tenengolts (VT) codes are a class of codes which can correct a single deletion or insertion with a linear-time decoder. This paper addresses the problem of efficient encoding of nonbinary VT codes, defined over an alphabet of size q > 2. We propose a simple linear-time encoding method to systematically map binary message sequences onto VT codewords. The method provides a new lower bound on the size of q-ary VT codes of length n. I. INTRODUCTIONDesigning codes for correcting deletions or insertions is well known to be a challenging problem; see, e.g., [1]- [8]. For the special case of correcting one insertion or deletion, there exists an elegant class of codes called Varshamov-Tenengolts (VT) codes. Binary VT codes were first introduced by Varshamov and Tenengolts in [9] for channels with asymmetric errors.Later, Levenshtein [10] showed that they can be used for correcting a single deletion or insertion with a simple decoding algorithm whose complexity is linear in the code length [1]. Tenengolts subsequently introduced a non-binary version of VT codes, defined over a q-ary alphabet for any q > 2 [11]. The q-ary VT codes retain many of the attractive properties of the binary codes. In particular, they can correct deletion or insertion of a single symbol from a q-ary VT codeword with a linear-time decoder.M. Abroshan and R. Venkataramanan are with the

Design of Nonbinary Error Correction Codes With a Maximum Run-Length Constraint to Correct a Single Insertion or Deletion Error for DNA Storage

“…The nonbinary VT codes [15], which also can correct a single insertion/deletion error, were proposed almost TABLE 1. The limitation and weakness of the related works [16], [18] and improvement of our work…”

“…Improvement of our work [16] • Code construct failure (CCF) issue was not considered, and some parity symbols could not be generated.…”

New Levenshtein-Marker Code for DNA-based Data Storage Capable of Correcting Multiple Edit Errors

“…Here = − is the number of redundancy bits. Many scholars have studied the generalization of VT code to non-binary alphabets [24], [25], [26].…”

“…Tenengolts's paper introduced the algebraic structure of the code, but didn't provide a method to encode messages into such codes. Only recently, Abroshan et al [25] proposed a method to systematically map bits message onto length -ary generalized Levenshtein codeword. We call it Encoder TA (introduced by Tenengolts and systematic encoded by Abroshan).…”

“…The Varshamovs-Tenengolts (VT) code, also called the Levenshtein code, is one of the most celebrated classes for correcting synchronization errors. VT codes that could correct single [22] or multiple [23] indels on binary alphabet have been designed, and also be extended to non-binary alphabets [24], [25], [26], with an open-source implementation available at [27]. Many generalized Levenshtein methods have been applied to the DNA storage system [20], [28], [29], [30], [31].…”