Additional Information Bits Research Articles

Generalized Precoding-Aided Quadrature Spatial Modulation

In this paper, we propose a novel scheme, which is called generalized precoding-aided quadrature spatial modulation (GPQSM), that extends the conventional quadrature spatial modulation to the receiver side. In GPQSM, spatial modulation works in both the in-phase and quadrature parts of the received signals, thus conveying additional information bits compared with conventional generalized precoding-aided spatial modulation (GPSM). The proposed scheme is general and can degenerate into the conventional multiple-input multiple-output (MIMO) scheme. A closed-form upper bound on the average bit error probability of GPQSM is derived. Simulation results verify the theoretical analysis and show that GPQSM outperforms the conventional GPSM scheme and the MIMO scheme under the same spectral efficiency.

Optical dual-mode index modulation aided OFDM for visible light communications

Dual-mode index modulation aided orthogonal frequency multiplexing (DM-OFDM) is recently proposed, where the spectral efficiency is enhanced compared with conventional OFDM schemes. In this paper, we propose two different optical DM-OFDM schemes for visible light communications, dual-mode index modulation aided DC-biased optical OFDM (DM-DCO-OFDM) and dual-mode index modulation aided unipolar OFDM (DM-U-OFDM). In the optical DM-OFDM schemes, subcarriers are partitioned into OFDM subblocks, divided into two groups within each subblock, and modulated by two different modes of constellations. Additional information bits can be transmitted implicitly by the indices of subcarriers modulated by the same constellation alphabet. In order to generate non-negative signals, the real-valued time-domain signals are DC-biased in DM-DCO-OFDM, whilst positive and negative signals are transmitted separately in DM-U-OFDM. At the receiver, both a maximum-likelihood (ML) detector and a log-likelihood ratio (LLR) detector are employed for signal demodulation. Besides, simulation results demonstrate that the proposed optical DM-OFDM schemes are capable of enhancing the spectral efficiency compared with other existing optical OFDM schemes, and DM-DCO-OFDM as well as DM-U-OFDM can achieve significant performance gains over their conventional counterparts at the same spectral efficiency, when the nonlinear transfer characteristic of light-emitting diodes (LEDs) is considered.

Vector OFDM With Index Modulation

The concept of index modulation (IM) has gained increased attention in recent years. As a member of the IM family, orthogonal frequency division multiplexing with index modulation (OFDM-IM), which is able to convey additional information bits using the indices of the active subcarriers, has emerged as an attractive physical layer transmission technique. Vector OFDM (V-OFDM) has inherently good peak-to-average-power ratio (PAPR) feature and good bit error rate (BER) performance due to the nearly full diversity gains achieved by most of its sub-blocks. In this paper, in order to improve the BER performance of OFDM-IM and reduce the PAPR of the transmit signal, we propose an enhanced OFDM-IM scheme termed vector OFDM with index modulation (V-OFDM-IM). In addition to the structure of the transceiver, a complexity reduced maximum likelihood detector is presented. To further reduce the detection complexity, a near-optimal detector based on sequential Monte Carlo technique is introduced. The average bit error probability of the proposed scheme is derived in closed form over frequency-selective block Rayleigh-fading channel. Both the theoretical and the simulation-based results show that, under the same spectrum efficiency, V-OFDM-IM outperforms the conventional OFDM-IM and OFDM.

Generalized Frequency Division Multiplexing With Flexible Index Modulation

After the vision and the overall objectives of future wireless networks for 2020 and beyond have been defined, standardization activities for fifth generation (5G) wireless networks have been started. Although it is expected that 5G new radio (NR) will be based on cyclicly prefixed orthogonal frequency division multiplexing (CP-OFDM)-based waveforms along with multiple waveform numerologies, the sufficiency of CP-OFDM-based NR is quite disputable due to the continuing massive growth trend in number of wireless devices and applications. Therefore, studies on novel radio access technologies (RATs) including advanced waveforms and more flexible radio accessing schemes must continue for future wireless networks. Generalized frequency division multiplexing (GFDM) is one of the prominent non-CP-OFDM-based waveforms. It has recently attracted significant attention in research because of its beneficial properties to fulfill the requirements of future wireless networks. Multiple-input multiple-output (MIMO)-friendliness is a key ability for a physical layer scheme to satisfactorily match the foreseen requirements of future wireless networks. On the other hand, the index modulation (IM) concept, which relies on conveying additional information bits through indices of certain transmit entities, is an emerging technique to provide better spectral and energy efficiency. In this paper, considering the advantages of non-CP-OFDM-based waveforms and the IM concept, we present a framework, which integrates GFDM with space and frequency IM schemes to provide flexible and advanced novel RATs for future wireless networks. Several MIMO-GFDM schemes are provided through the proposed framework and their bit error ratio performances, computational complexities, and spectral efficiencies are analyzed. Based on the obtained results, a guideline for selecting the proper MIMO-GFDM scheme considering target performance criterion is given. It has been demonstrated that the proposed framework has a strong potential to engineer the space-frequency structure according to channel conditions and use cases, and it provides a great flexibility that can be easily tuned to address the required performance criterion.

Error Probability Analysis of OFDM-IM With Carrier Frequency Offset

OFDM-IM, which uses the indices of active subcarriers to carry additional information bits, is a novel multi-carrier transmission technique. In this letter, by exploiting the special design of OFDM-IM, we propose an analytical approach to evaluate the error probability of OFDM-IM subject to carrier frequency offset (CFO) in fading channels, where the inter-carrier interference (ICI) is properly divided into two parts: the intra-subblock ICI and the inter-subblocks ICI. Simulation results illustrate that the results of theoretical analysis are quite accurate, and OFDM-IM outperforms the conventional OFDM when CFO exists.

Equiprobable Subcarrier Activation Method for OFDM With Index Modulation

Orthogonal frequency division multiplexing with index modulation (OFDM-IM) conveys additional information bits via the indices of active subcarriers. Consequently, the determination of the active subcarriers according to the incoming bits arises as a challenging problem. The existing solution resorts to the lexicographic ordering, which leads to a low implementation complexity but an unequal subcarrier activation probability. This letter proposes a distinct solution that allows the equiprobable activation of all OFDM-IM subcarriers with comparable implementation complexity. The signal-to-noise ratio gain achieved by the proposed solution is also analyzed. Computer simulations reveal the advantages of the proposed solution.

Reliable data transfer Rendezvous protocol in wireless sensor networks using 2D-SEC-DED encoding technique

Many applications in Wireless Sensor Networks (WSNs) require all data to be transmitted with minimal or without loss, what implies that reliability is an important characteristic. In any WSN, there are two basic approaches to recover erroneous packets. One way is to use Automatic Repeat reQuest (ARQ), and another is Forward Error Correction (FEC). The error-control systems for applications based on ARQ use error detection coupled with retransmission requests to maximize reliability at some cost to throughput. Error detection is generally provided by the lower protocol layers which use checksums (e.g. Cyclic Redundancy Checksums (CRCs)) to discard corrupted packets and trigger retransmission requests. In these solutions event a single erroneous bit can render a packet useless to the end user. Having in mind that in WSNs the power is scarce and is primarily consumed by wireless transmission and reception, we propose to use FEC rather than ARQ. FEC is a way of correcting packets by transmitting additional information bits with aim to reduce the frequency of retransmission requests. During this, data bytes are optionally encoded after being fragmented with Error Correcting Code (ECC) to recover data bits in case of small number of bit errors. Various FEC encoding schemes such as erasure and Hamming based codes are available. The choice of the encoding schemes depends on the applications and error characteristics (error models/patterns) of the wireless channel. Erasure encoding is preferable for usage when the error pattern is burst dominated, while Hamming encoding when noise causes random errors. Our observations show that most bit errors are single-bit or double-bit errors and burst errors are present but rare. In this work, an efficient Hamming based FEC encoding scheme of relatively low complexity called Two Dimensional-Single Error Correcting-Double Error Detecting (2D-SEC-DED), intended to minimize packet retransmissions and to save energy, has been developed. Such FEC scheme can be used to correct all single-bit and 99.99%of double/multiple-bit errors. Since the radio block is dominant energy consumer within a Sensor Node (SN), we focus our attention to answer the question: which is the adequate metric to use, and under what conditions to accurately characterize the quality of the communication, related to reliable data transfer, with minimal energy consumption. To this end, as first, in a case when the bit error is not high and most errors are single-bit, we show that 2D SEC-DED encoding scheme is more energy efficient in comparison to erasure encoding. As second, the advantages of using 2D-SEC-DED in respect to CRC (NO-FEC) encoding, concerning decreasing the energy consumption and increasing the reliability of the radio block are derived through implementation of two versions of the Rendezvous Protocol for Long Living (RPLL) referred as Modified-RPLL (M-RPLL as FEC based) and Ordinary-RPLL (O-RPLL as NO-FEC), respectively.

SM/SPPM Aided Multiuser Precoded Visible Light Communication Systems

In this paper, we propose a multiuser precoded visible light communication (VLC) system utilizing optical orthogonal frequency division multiplexing (OFDM), where optical spatial modulation (OSM) and spatial pulse position modulation (SPPM) are invoked for supporting high-rate data transmissions without compromising uniform illumination across multiple light-emitting diodes (LEDs). Both OSM and SPPM exploit the index of data streams for carrying additional information bits, whereas the multiuser interference occurring between different user equipments is mitigated by the MIMO precoder. By evaluating the performances of the proposed VLC systems, we demonstrate that the OSM-aided system is capable of achieving high data rates at sufficiently high signal-to-noise ratios (SNRs). In contrast, the SPPM-aided system has a higher effective bandwidth efficiency at low SNRs, despite the fact that its attainable peak data rate is lower than that of the SM-aided counterpart. Such tradeoffs indicate the potential application scenarios of the proposed schemes for achieving diverse design prerequisites.

Low-Complexity ML Detector and Performance Analysis for OFDM With In-Phase/Quadrature Index Modulation

A novel variant of orthogonal frequency division multiplexing (OFDM) techniques, which carries additional information bits through the index domain including in-phase and quadrature dimensions, is recently proposed. For its nature, we refer to this technique as OFDM with in-phase/quadrature index modulation (OFDM-I/Q-IM) . In this letter, we propose a novel low-complexity detector based on the maximum-likelihood (ML) criterion for OFDM-I/Q-IM, which does not need to know the variance of the noise and the possible realizations of the active subcarrier indices. With the proposed ML detector, the asymptotic average bit error probability (ABEP) and the exact coding gain achieved by OFDM-I/Q-IM are also derived.

Generalization of Orthogonal Frequency Division Multiplexing With Index Modulation

Recently, orthogonal frequency division multiplexing (OFDM) with index modulation (OFDM-IM) was proposed. By selecting a fixed number of subcarriers as active subcarriers to carry constellation symbols, the indices of these active subcarriers may carry additional bits of information. In this paper, we propose two generalization schemes of OFDM-IM, named OFDM with generalized index modulation 1 (OFDM-GIM1) and OFDM-GIM2, respectively. In OFDM-GIM1, the number of active subcarriers in an OFDM subblock is no longer fixed. Dependent on the input binary string, different numbers of active subcarriers are assigned to carry constellation symbols. In OFDM-GIM2, independent index modulation is performed on the in-phase and quadrature component per subcarrier. Through such ways, a higher spectral efficiency than that of OFDM-IM may be achieved. Since both generalization schemes proposed suffer from BER performance loss in low SNR region, an interleaving technique is proposed to tackle this problem. Finally, noting that the two generalization schemes are compatible with each other, the combination of these two schemes, named OFDM-GIM3, has also been investigated. Computer simulation results clearly show our proposed scheme's superiority in both spectral efficiency and BER performance compared to existing works.

Optimal encoding capacity of a linear optical quantum channel

Here, we study the capacity of a quantum channel, assuming linear optical encoding, as a function of available photons and optical modes. First, we observe that substantial improvement is made possible by not restricting ourselves to a rail-encoded qubit basis. Then, we derive an analytic formula for general channel capacity and show that this capacity is achieved without requiring the use of entangling operations typically required for scalable universal quantum computation, e.g. KLM measurement-assisted transformations. As an example, we provide an explicit encoding scheme using the resources required of standard dense coding using two dual-rail qubits (2 photons in 4 modes). In this case, our protocol encodes one additional bit of information. Greater gains are expected for larger systems.

Generalized Space-and-Frequency Index Modulation

Unlike in conventional modulation where information bits are conveyed only through symbols from modulation alphabets defined in the complex plane (e.g., quadrature amplitude modulation (QAM), phase shift keying (PSK)), in index modulation (IM), additional information bits are conveyed through indices of certain transmit entities that get involved in the transmission. Transmit antennas in multi-antenna systems and subcarriers in multi-carrier systems are examples of such transmit entities that can be used to convey additional information bits through indexing. In this paper, we introduce {\em generalized space and frequency index modulation}, where the indices of active transmit antennas and subcarriers convey information bits. We first introduce index modulation in the spatial domain, referred to as generalized spatial index modulation (GSIM). For GSIM, where bits are indexed only in the spatial domain, we derive the expression for achievable rate as well as easy-to-compute upper and lower bounds on this rate. We show that the achievable rate in GSIM can be more than that in spatial multiplexing, and analytically establish the condition under which this can happen. It is noted that GSIM achieves this higher rate using fewer transmit radio frequency (RF) chains compared to spatial multiplexing. We also propose a Gibbs sampling based detection algorithm for GSIM and show that GSIM can achieve better bit error rate (BER) performance than spatial multiplexing. For generalized space-frequency index modulation (GSFIM), where bits are encoded through indexing in both active antennas as well as subcarriers, we derive the achievable rate expression. Numerical results show that GSFIM can achieve higher rates compared to conventional MIMO-OFDM. Also, BER results show the potential for GSFIM performing better than MIMO-OFDM.

Finding Miss Dilettant: Rose Hawthorne Lathrop's Recovered Novel

Rose Hawthorne's only complete novel exists in the Boston Public Library's microfilm department. Reference librarians at the Library of Congress and at the Boston Public Library helped me recover, Miss Dilettant, (1) Rose Hawthorne Lathrop's only published novel. I mention it now to include it in the Rose Hawthorne Lathrop chronology published in the Nathaniel Hawthorne Review (Fall 2010). Partially completed novels by Rose exist in the Rosary Hill archives, but Miss Dilettant (1879) is Rose's only recovered complete novel. Interestingly, it looks back to The Scarlet Letter with Hawthorne's characterization of Hester as "self-ordained a Sister of Mercy" (2) and forward with Rose's characterization of her heroine's first encounter serving the poor of New York as a "fresh sister-of-mercy." (3) Both characterizations foreshadow Rose's later work with destitute cancer patients. George Lathrop, Rose's husband and editor of the Boston Courier (1877-1879), published Miss Dilettant, almost 70,000 words and eighteen chapters, in sixteen installments in the newspaper's weekend edition from 5 April to 29 July 1879. George's assistance in the publication and preservation of Miss Dilettant cannot be discounted. One may safely say he was the novel's first editor. With no manuscript currently available, the Boston Courier and the Boston Public Library protected the work for two centuries--no mean service for an editor, a newspaper, and a library. With increased literacy, improved printing capability, and better distribution procedures, serialized fiction gained acceptance in England during the Victorian period (1837-1901) about the time Charles Dickens, George Eliot, and William Thackeray for economic reasons selected installment publication for their new novels. This literary trend caught hold in the United States where serial novels, printed inexpensively, increased newspaper and journal circulation by readers eager for the next episode. Scribner's, Harper's, and the Atlantic Monthly gained credibility as serious serial periodicals, offered readers the best of America's literary talent, and provided incomes for writers. Aware of literary trends and economic reality, editor George Lathrop increased weekend circulation for the Boston Courier when he placed his wife's novel in the mainstream of publication practice. Gradually more American writers preferred to offer their work in installments before it appeared in book form. Rose gave birth to Francis Hawthorne Lathrop 10 November 1876, but she spent months in recovery from post-partum depression, returning home from hospitalization in early 1877 with joy in motherhood that stirred the first chapters of a novel during that year. (4) An early biographer, Theodore Maynard, in A Fire Was Lighted dismissed the existence of a finished book, noting that Rose "made one of her many incompleted attempts to produce a novel." (5) He added that she received a letter from Thomas Niles 28 December 1877 of the Boston publishers Robert Brothers who had read the first seven chapters of a novel, "rather noncommittally." In truth, Niles wrote on that very date that "I have read with much interest" the first seven chapters and have "been expecting the balance of Miss Dilletante." His spelling of "Dilletante" is significant in that it allows readers to trace where biographers discovered the novel and indicates that the title's spelling changed from the time he read Rose's draft to its actual publication. He added, "The character drawing, especially of the heroine, is excellent--the style original but with a peculiarity which, I think, obscures the meaning." Niles concluded, "It is not fair, however, to say much about an unfinished work which I am quite anxious to see the whole of." (6) Niles's letter gives an additional bit of information: by 28 December 1877, Rose had completed seven chapters of a novel that would ultimately comprise eighteen chapters by March 1879. Maynard evidently did not pursue Miss Dilettant, because no manuscript exists in the Rosary Hill archives whose records helped him write his biography. …

A History of Failure

Blonde Cobra begins with silence and a still black-and-white image. After several seconds, a voice asks, are your favorite Gershwin songs? This voice seems to prompt movement within the image, as well as the sound of a second voice quickly rattling off a list of songs in response: I've Got Rhythm, Liza, S'Wonderful, . . . and Thee Sing.1 first voice interrupts the second's recitation with the observation that most of these songs are in the movies as the second voice continues to suggest a few more songs. Then the conversation turns to Victor Moore, an actor in the Broadway musical Of Thee Sing (director George S. Kaufman, 1931).First voice: Victor Moore is dead.Second voice: Yeah. I'm sorry.First: Who'd he play? What was the part he played? Ta ta tum ta tuttlebaum . . . Tuttlebaum. Tuttlebaum. Throttlebottom.Second: What? What was Throttlebottom's first name?First: How did the song go?Second: was his first name, Throttlebottom? Faster, Robert faster.First: I give up.Second: Alexander.First: Alexander Throttlebottom.Second: part, position did he have, did he play in?First: The Vice President.Second: Correct Robert.This rapid exchange continues for a few more seconds with the two voices asking and answering questions or offering each other additional bits of information about Of Thee Sing, including its date-1931-and the fact that it won the Pulitzer Prize, didn't it? Then, after repeating an earlier question-How did the songs in it go?-the second voice begins to whistle, and both voices then vocalize the tune and ultimately some of the lyrics to one of the songs from the musical: Bum bum bum bum bumbabum . . . Wintergreen for President, da dada dadadada. And then the first voice asks, What's the other one? voice of Ginger Rogers singing the title from Let's Call the Whole Thing Off provides a response, the figure on the screen moves toward the camera and quickly fades out of focus, handwritten numbers indicating the end of a reel appear, and then the screen goes black for quite a while, long enough to tempt uninitiated viewers into thinking that maybe this last line was meant to signal the end of the film or at least to potentially describe an attitude toward the film project as a whole: a collaboration doomed to failure.2Blonde Cobra did not, in fact, come easily. At a number of points, it came dangerously close to being abandoned because of severely strained personal relationships among its collaborators. Bob Fleischner shot the footage for what he and Jack Smith initially intended to be two light monster comedies during the winter of 1959. Soon afterward, a fire in Smith's apartment destroyed some raw stock that Fleischner had been storing there. Fleischner insisted that Smith repay him for the ruined stock, which was an unrealistic demand since the cause of the fire, Smith's cat knocking over a candle, was indicative of the fact that Smith had no money. Con Edison had already turned off his electricity for nonpayment. Smith and Fleischner had a falling out, and Fleischner gave the unedited footage-one color and ten black-and-white, silent, hundred-foot 16-mm rolls-to another filmmaker, Ken Jacobs, in the hopes that he could salvage something from it.3 Jacobs claimed that his ignorance of Smith and Fleischner's original intentions for the project provided a distinct advantage: Having no idea of the original story plans was able to view the material, not as exquisite fragments of a failure, of two failures, but as the makings of a new entity. Bob gave over the footage to me and with it the freedom to develop it as saw fit.4 Jacobs edited the footage, adding some short color sequences that he had previously shot of Fleischner, Smith, and Jerry Sims, who also appears in the original Blonde Cobra footage, and a soundtrack, completing the film in 1963. …

A Set of Space-Time Block Codes for the High-Rate Transmission Scheme with One Information Bit

In [1], the authors have proposed one high rate transmission scheme for Space-Time Block Codes (STBC) without additional system source such as power, bandwidth and time slot. To maintain the full rank property of the coding gain matrix, we propose a set of STBCs for 4 transmit antennas transmission to transmit one additional information bit achieving rate-9/8. Another orthogonal STBC code with rate-1 is proposed in this paper within the set. It shows by computer simulation results that by employing the set of STBCs, it achieves better bit error rate (BER) performance and throughput than that of [1] with a valid BER improvement at the high SNR region above 20dB.

A High Rate Transmission for Space-Time Block Coding with One Additional Bit Information

In this letter, we propose a set of Space-Time Block Codes (STBC) equipped with 4 transmit antennas to transmit one additional information bit achieving rate-9/8. To maintain full rank property of the coding gain matrix, one new orthogonal STBC code with full rate is proposed in this letter. Simulation results show that this method achieves good bit error rate (BER) performance with a small gap compared to that of the rate-1 case and throughput in the high SNR region.

Robust Image Transmission Over Wireless Sensor Networks

Robust image and video communications have become more imperative due to the ubiquitous proliferation of multimedia applications over wireless sensor networks. In this work, the transmission distortions on the image data induced by both channel and instant node failures for Wireless Sensor Networks (WSN) are considered. The effect of two techniques and their integration with multipath transmission are investigated to compensate the multimedia distortions at the expense of incurring additional energy consumption and/or wasting bandwidth resources. First technique is watermarking based error concealment utilizing discrete wavelet transform for embedding downsized replicas of original image into itself. The other is conventional Reed---Solomon (RS) coding utilizing additional information bits to correct bit/symbol errors. Performance results obtained from extensive simulations utilizing a communication and energy model applicable to WSN show that error concealment (EC) integrated schemes, especially EC with multipath fusion (ECMF), are more promising to compensate losses than RS-coding-integrated and pure multipath transmission techniques in WSN.

Error Resilient LZ'77 Data Compression: Algorithms, Analysis, and Experiments

We propose a joint source-channel coding algorithm capable of correcting some errors in the popular Lempel-Ziv'77 (LZ'77) scheme without introducing any measurable degradation in the compression performance. This can be achieved because the LZ'77 encoder does not completely eliminate the redundancy present in the input sequence. One source of redundancy can be observed when an LZ'77 phrase has multiple matches. In this case, LZ'77 can issue a pointer to any of those matches, and a particular choice carries some additional bits of information. We call a scheme with embedded redundant information the LZS'77 algorithm. We analyze the number of longest matches in such a scheme and prove that it follows the logarithmic series distribution with mean 1/h (plus some fluctuations), where h is the source entropy. Thus, the distribution associated with the number of redundant bits is well concentrated around its mean, a highly desirable property for error correction. These analytic results are proved by a combination of combinatorial, probabilistic, and analytic methods (e.g., Mellin transform, depoissonization, combinatorics on words). In fact, we analyze LZS'77 by studying the multiplicity matching parameter in a suffix tree, which in turn is analyzed via comparison to its independent version, called trie. Finally, we present an algorithm in which a channel coder (e.g., Reed-Solomon (RS) coder) succinctly uses the inherent additional redundancy left by the LZS'77 encoder to detect and correct a limited number of errors. We call such a scheme the LZRS'77 algorithm. LZRS'77 is perfectly backward-compatible with LZ'77, that is, a file compressed with our error-resistant LZRS'77 can still be decompressed by a generic LZ'77 decoder

Small Accessible Quantum Information Does Not Imply Security

The security of quantum key distribution is typically defined in terms of the mutual information between the distributed key S and the outcome of an optimal measurement applied to the adversary's system. We show that even if this so-called accessible information is small, the key S might not be secure enough to be used in applications such as one-time pad encryption. This flaw is due to a locking property of the accessible information: one additional (physical) bit of information can increase the accessible information by more than one bit.

Informational Complexity and Functional Activity of RNA Structures

Very little is known about the distribution of functional DNA, RNA, and protein molecules in sequence space. The question of how the number and complexity of distinct solutions to a particular biochemical problem varies with activity is an important aspect of this general problem. Here we present a comparison of the structures and activities of eleven distinct GTP-binding RNAs (aptamers). By experimentally measuring the amount of information required to specify each optimal binding structure, we show that defining a structure capable of 10-fold tighter binding requires approximately 10 additional bits of information. This increase in information content is equivalent to specifying the identity of five additional nucleotide positions and corresponds to an ∼1000-fold decrease in abundance in a sample of random sequences. We observe a similar relationship between structural complexity and activity in a comparison of two catalytic RNAs (ribozyme ligases), raising the possibility of a general relationship between the complexity of RNA structures and their functional activity. Describing how information varies with activity in other heteropolymers, both biological and synthetic, may lead to an objective means of comparing their functional properties. This approach could be useful in predicting the functional utility of novel heteropolymers.