Bits Of Redundancy Research Articles

Repeat-Free Codes

In this paper we consider the problem of encoding data into repeat-free sequences in which sequences are imposed to contain any k-tuple at most once (for predefined k). First, the capacity of the repeat-free constraint are calculated. Then, an efficient algorithm, which uses two bits of redundancy, is presented to encode length- n sequences for k=2+2log(n). This algorithm is then improved to support any value of k of the form k=alog(n), for 1 <; a, while its redundancy is o(n). We also calculate the capacity of repeat-free sequences when combined with local constraints which are given by a constrained system, and the capacity of multi-dimensional repeat-free codes.

Coded Trace Reconstruction

Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of coded trace reconstruction , the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called traces ) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i.d. deletions and constant deletion probability. Our work is the first step towards the design of efficient codes with provable guarantees for such systems. We consider a constant rate of i.i.d. deletions, and perform an analysis of marker-based code-constructions. This gives rise to codes with redundancy $O(n/\log n)$ (resp. $O(n/\log \log n)$ ) that can be efficiently reconstructed from $\exp (O(\log ^{2/3}n))$ (resp. $\exp (O(\log \log n)^{2/3})$ ) traces, where $n$ is the message length. Then, we give a construction of a code with $O(\log n)$ bits of redundancy that can be efficiently reconstructed from $\text {poly}(n)$ traces if the deletion probability is small enough. Finally, we show how to combine both approaches, giving rise to an efficient code with $O(n/\log n)$ bits of redundancy which can be reconstructed from $\text {poly}(\log n)$ traces for a small constant deletion probability.

Codes Correcting Two Deletions

In this paper, we investigate the problem of constructing codes capable of correcting two deletions. In particular, we construct a code that requires redundancy approximately 8 log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n + O(log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n) bits of redundancy, where n denotes the length of the code. To the best of the authors' knowledge, this represents the best known construction in that it requires the lowest number of redundant bits for a code correcting two deletions.

Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from $k$ –bit deletions with efficient encoding/decoding, for a fixed $k$ . The single deletion case is well understood, with the Varshamov–Tenengolts–Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx ~2^{n}/n$ codewords of length $n$ , i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{\Omega (1)}$ . For any fixed $k$ , we construct a binary code with $c_{k} \log n$ redundancy that can be decoded from $k$ deletions in $O_{k}(n \log ^{4} n)$ time. The coefficient $c_{k}$ can be taken to be $O(k^{2} \log k)$ , which is only quadratically worse than the optimal, non-constructive bound of $O(k)$ . We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions. We also note that among linear codes capable of correcting $k$ deletions, the $(k+1)$ -fold repetition code is essentially the best possible.

A 1.7 mW 11b 250 MS/s 2-Times Interleaved Fully Dynamic Pipelined SAR ADC in 40 nm Digital CMOS

A 250 MS/s 2x interleaved 11 bit pipelined SAR ADC in 40 nm digital CMOS is presented. Each ADC channel consists of a 6b coarse SAR, a dynamic residue amplifier and a 7b fine SAR with a total of two bits of redundancy. Calibration is leveraged to adjust the uncertain gain of the chosen residue amplifier and various other non-idealities. The ADC achieves a peak SNDR of 62 dB at 10 MS/s, and 56 dB for a Nyquist input at 250 MS/s. The low frequency energy per conversion step ranges from 7 fJ at 10 MS/s to 10 fJ at 250 MS/s.

Encoding finite sources with ranked probabilities

The problem of designing a code for a source with at most N symbols and ranked probabilities is investigated. The alphabet size is considered as a random variable which takes values in { 1 , 2 , … , N } with probability 1 / N . The average distribution whose associated Huffman code is the best choice, according to the Minimum Average criterion, is derived. Some properties of the average distribution are studied and it is shown that a simple fixed-length code with codewords of length ⌈ log N ⌉ has at most 2.67 bits of additional redundancy with respect to the optimum code. A simple and efficient suboptimal code is proposed whose redundancy is less than 0.2 for N ⩾ 7 . It is shown that the fixed-length code is a suitable code even when small (or large) alphabet sizes are more probable.

A Balancing Method for PMEPR Reduction in OFDM Signals

A relation is established between the strength of a binary code over the alphabet {+1,-1}, and its ability to reduce peak-to-mean envelope power ratio (PMEPR) in n-subcarrier (OFDM) signals. Based on this relation, a method is proposed to deterministically bound PMEPR of such signals using coordinate-wise multiplication by a balancing vector (BV) chosen from a code of given strength. A practical probabilistic scheme considering a small number of candidate codewords is devised. For this scheme, estimates on the PMEPR reduction achievable with arbitrary high probability are derived. In particular, the scheme provides for large n PMEPR of lnn+2.01lnlnn with (ln2)middot(log2n)2+1 bits of redundancy, the failure probability at most e-n, and testing n/(lnlnn) candidate BVs. Finally, several practical settings are considered. For example, for quaternary phase-shift keying, n=128, the scheme with 36 bits of redundancy (18 redundant subcarriers), by testing only 4 BVs provides over 2 dB PMEPR reduction, for any failure rate below 10-2.5

Asymptotic limits in peak envelope power reduction by redundant coding in orthogonal frequency-division multiplex modulation

This paper uses some computational results to show that the peak envelope power (PEP) of orthogonal frequency-division multiplex (OFDM) waveforms can be reduced very substantially by coding to select only those messages with low peak factor as valid codewords in the scheme. It is shown that the asymptotic limit of the PEP is between two and three, and that only four bits of redundancy are necessary to limit the PEP to within 10% of this optimum value as the number of carriers becomes large.

The distribution of peak factor in QPSK Multi-Carrier modulation

This paper presents the results of a preliminary investigation into the distribution of peak factors in the QPSK message space for different numbers of carriers with the aim of determining those messages which exhibit a low peak factor relative to a given threshold. A group structure is suggested which associatescosets of 16 messages intoequivalence classes having the same peak power. Algorithms for finding the members of cosets and equivalence classes are given. These messages are then rank ordered by peak power to show how a suitable code could be derived. It appears that allocating just one or two bits of redundancy to such a code will be sufficient to achieve substantial reductions in peak factor.

A method for updating a cyclic redundancy code

The contents of error-protected frames can be intentionally altered as the frames traverse through a digital network. The check bits must be recomputed to conform with the altered text. A method is described for updating the check bits of a cyclic redundancy check (CRC) code, based on knowledge of the altered bits and their position in the frame. Unlike previous methods, its complexity is independent of the frame size. >

Performance of Error-Correcting Codes

Binary decoding of relatively short codes corrupted by Gaussian noise or pulse interference has been considered in applications for which the minimum useful information block length is unity, equal to the information bit capacity of a code, or equal to an integral multiple of the information capacity of a code. The following results pertain to applications which are further characterized by the performance criterion of minimizing error rate and transmitter power, with only minor restrictions on transmission bandwidth and none on channel efficiency (no credit given for error detection): [1] The word error probability for any group error-correcting code is never exceeded by its corresponding bit error probability (for either Gaussian noise or pulse interference). [2] Relatively short codes (10 to 50 information bits) of moderate redundancy (40 to 50 per cent) are available which can reduce bit error rates by factors of 10 to 400, or equivalently, permit 1 to 3 db reductions in transmitter power, when the coded system bit error rate is 10-6(Gaussian noise). [3] These same codes can alternatively reduce word error rates by factors of 10 to 104, or equivalently, permit reductions of 1.5 to 3.7 db in transmitter power, for average word error rates corresponding to values of the quantity E/N_{o} (signal energy per information bit/Gaussian noise power density) in the neighborhood of 8 db for the coded systems. [4] Under worst-case pulse interference, these codes permit 4 to 7 db reductions in transmitter power for average word error rates less than 20 per cent. (Considerably greater improvements result for the pulse interference expected.) [5] All of these codes are easily encoded and decoded.