Yaakov Ziv

Yaakov Ziv is an Israeli electrical engineer who, along with Abraham Lempel, developed the LZ family of lossless data compression algorithms. Ziv was born in Tiberias, British-ruled Palestine, on 27 November 1931, he received the B. Sc. Dip. Eng. and M. Sc. degrees, all in electrical engineering, from the Technion – Israel Institute of Technology in 1954, 1957 and the D. Sc. degree from the Massachusetts Institute of Technology in 1962. Ziv joined the Technion – Israel Institute of Technology in 1970 and is Herman Gross Professor of Electrical Engineering and a Technion Distinguished Professor, his research interests include data compression, information theory, statistical communication theory. Ziv was Dean of the Faculty of Electrical Engineering from 1974 to 1976 and Vice President for Academic Affairs from 1978 to 1982. Since 1987 Ziv has spent three sabbatical leaves at the Information Research Department of Bell Laboratories in Murray Hill, New Jersey, USA. From 1955 to 1959, he was a Senior Research Engineer in the Scientific Department Israel Ministry of Defense, was assigned to the research and development of communication systems.

From 1961 to 1962, while studying for his doctorate at M. I. T, he joined the Applied Science Division of Inc.. Watertown, MA, where he was a Senior Research Engineer doing research in communication theory. In 1962 he returned to the Scientific Department, Israel Ministry of Defense, as Head of the Communications Division and was an Adjunct of the Faculty of Electrical Engineering, Technion - Israel Institute of Technology. From 1968 to 1970 he was a Member of the Technical Staff of Inc.. Ziv was the Chairman of the Israeli Universities Planning and Grants Committee from 1985 to 1991, he has been a member of the Israel Academy of Sciences and Humanities since 1981 and served as its president between 1995 and 2004. In 1993, Ziv was awarded the Israel Prize, for exact sciences. Ziv received in 1995 the IEEE Richard W. Hamming Medal, for "contributions to information theory, the theory and practice of data compression", in 1998 a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society.

Ziv is the recipient of the 1997 Claude E. Shannon Award from the IEEE Information Theory Society and the 2008 BBVA Foundation Frontiers of Knowledge Award in the category of Information and Communication Technologies; these prestigious awards are considered second only to the Nobel Prize in their monetary amount. List of Israel Prize recipients A Conversation with Jacob Ziv ACM Paris Kanellakis Theory and Practice Award 1977: Jacob Ziv Jacob Ziv at DBLP Bibliography Server

Mathematics

Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.

The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to

Moscow State University

Moscow State University is a coeducational and public research university located in Moscow, Russia. It was founded on 23 January 1755 by Mikhail Lomonosov. MSU was renamed after Lomonosov in 1940 and was known as Lomonosov University, it houses the tallest educational building in the world. Its current rector is Viktor Sadovnichiy. According to the 2018 QS World University Rankings, it is the highest-ranking Russian educational institution and is considered the most prestigious university in the former Soviet Union. Ivan Shuvalov and Mikhail Lomonosov promoted the idea of a university in Moscow, Russian Empress Elizabeth decreed its establishment on 23 January 1755; the first lectures were given on 7 May. Russians still celebrate 25 January as Students' Day. Saint Petersburg State University and Moscow State University engage in friendly rivalry over the title of Russia's oldest university. Though Moscow State University was founded in 1755, its competitor in St. Petersburg has had a continuous existence as a "university" since 1819 and sees itself as the successor of an academy established on 24 January 1724, by a decree of Peter the Great.

The present Moscow State University occupied the Principal Medicine Store on Red Square from 1755 to 1787. Catherine the Great transferred the University to a Neoclassical building on the other side of Mokhovaya Street. In the 18th century, the University had three departments: philosophy and law. A preparatory college was affiliated with the University until its abolition in 1812. In 1779, Mikhail Kheraskov founded a boarding school for noblemen which in 1830 became a gymnasium for the Russian nobility; the university press, run by Nikolay Novikov in the 1780s, published the most popular newspaper in Imperial Russia: Moskovskie Vedomosti. In 1804, medical education split into clinical and obstetrics faculties. During 1884–1897, the Department of Medicine—supported by private donations, the municipal and imperial governments—built an extensive, 1.6-kilometer-long, state-of-the-art medical campus in Devichye Pole, between the Garden Ring and Novodevichy Convent. The campus, medical education in general, were separated from the Moscow University in 1930.

Devichye Pole was operated by the independent I. M. Sechenov First Moscow State Medical University and by various other state and private institutions; the roots of student unrest in the University reach deep into the nineteenth century. In 1905, a social-democratic organization emerged at the University and called for the overthrow of the Czarist government and the establishment of a republic in Russia; the imperial government threatened to close the University. In 1911, in a protest over the introduction of troops onto the campus and mistreatment of certain professors, 130 scientists and professors resigned en masse, including such prominent men as Nikolay Dimitrievich Zelinskiy, Pyotr Nikolaevich Lebedev, Sergei Alekseevich Chaplygin. After the October Revolution of 1917, the institution began to admit the children of the proletariat and peasantry. In 1919, the University abolished fees for tuition and established a preparatory facility to help working-class children prepare for entrance examinations.

During the implementation of Joseph Stalin's first five-year plan, prisoners from the Gulag were forced to construct parts of the newly expanded University. After 1991, nine new faculties were established; the following year, the University gained a unique status: it is funded directly from the state budget, thus providing the University a significant level of independence. On 6 September 1997, the French electronic musician Jean Michel Jarre, whom the mayor of Moscow had specially invited to perform, used the entire front facade of the University as the backdrop for a concert: the frontage served as a giant projection screen, with fireworks and searchlights all launched from various points around the building; the stage stood directly in front of the building, the concert, entitled "The Road To The 21st Century" in Russia but renamed "Oxygen In Moscow" for worldwide release in video/DVD, attracted a world-record crowd of 3.5 million people. On 19 March 2008, Russia's most powerful supercomputer to date, the SKIF MSU was launched at the University.

Its peak performance of 60 TFLOPS makes it the fastest supercomputer in the Commonwealth of Independent States. Since 1953, most of the faculties have been situated on Sparrow Hills, in the southwest of Moscow, 5 km from the city centre; the main building was designed by architect Lev Vladimirovich Rudnev. In the post-war era, Joseph Stalin ordered seven huge tiered neoclassic towers to be built around the city, it was built using Gulag labour. Located on Moscow's outskirts at the time of its construction, the location of the main building is now about half-way between the center of Moscow a

Soviet Union

The Soviet Union the Union of Soviet Socialist Republics, was a socialist state in Eurasia that existed from 1922 to 1991. Nominally a union of multiple national Soviet republics, its government and economy were centralized; the country was a one-party state, governed by the Communist Party with Moscow as its capital in its largest republic, the Russian Soviet Federative Socialist Republic. Other major urban centres were Leningrad, Minsk, Alma-Ata, Novosibirsk, it spanned over 10,000 kilometres east to west across 11 time zones, over 7,200 kilometres north to south. It had five climate zones: tundra, steppes and mountains; the Soviet Union had its roots in the 1917 October Revolution, when the Bolsheviks, led by Vladimir Lenin, overthrew the Russian Provisional Government which had replaced Tsar Nicholas II during World War I. In 1922, the Soviet Union was formed by a treaty which legalized the unification of the Russian, Transcaucasian and Byelorussian republics that had occurred from 1918. Following Lenin's death in 1924 and a brief power struggle, Joseph Stalin came to power in the mid-1920s.

Stalin committed the state's ideology to Marxism–Leninism and constructed a command economy which led to a period of rapid industrialization and collectivization. During his rule, political paranoia fermented and the Great Purge removed Stalin's opponents within and outside of the party via arbitrary arrests and persecutions of many people, resulting in at least 600,000 deaths. In 1933, a major famine struck the country. Before the start of World War II in 1939, the Soviets signed the Molotov–Ribbentrop Pact, agreeing to non-aggression with Nazi Germany, after which the USSR invaded Poland on 17 September 1939. In June 1941, Germany broke the pact and invaded the Soviet Union, opening the largest and bloodiest theatre of war in history. Soviet war casualties accounted for the highest proportion of the conflict in the effort of acquiring the upper hand over Axis forces at intense battles such as Stalingrad and Kursk; the territories overtaken by the Red Army became satellite states of the Soviet Union.

The post-war division of Europe into capitalist and communist halves would lead to increased tensions with the United States-led Western Bloc, known as the Cold War. Stalin died in 1953 and was succeeded by Nikita Khrushchev, who in 1956 denounced Stalin and began the de-Stalinization; the Cuban Missile Crisis occurred during Khrushchev's rule, among the many factors that led to his downfall in 1964. In the early 1970s, there was a brief détente of relations with the United States, but tensions resumed with the Soviet–Afghan War in 1979. In 1985, the last Soviet premier, Mikhail Gorbachev, sought to reform and liberalize the economy through his policies of glasnost and perestroika, which caused political instability. In 1989, Soviet satellite states in Eastern Europe overthrew their respective communist governments; as part of an attempt to prevent the country's dissolution due to rising nationalist and separatist movements, a referendum was held in March 1991, boycotted by some republics, that resulted in a majority of participating citizens voting in favor of preserving the union as a renewed federation.

Gorbachev's power was diminished after Russian President Boris Yeltsin's high-profile role in facing down a coup d'état attempted by Communist Party hardliners. In late 1991, Gorbachev resigned and the Supreme Soviet of the Soviet Union met and formally dissolved the Soviet Union; the remaining 12 constituent republics emerged as independent post-Soviet states, with the Russian Federation—formerly the Russian SFSR—assuming the Soviet Union's rights and obligations and being recognized as the successor state. The Soviet Union was a powerhouse of many significant technological achievements and innovations of the 20th century, including the world's first human-made satellite, the first humans in space and the first probe to land on another planet, Venus; the country had the largest standing military in the world. The Soviet Union was recognized as one of the five nuclear weapons states and possessed the largest stockpile of weapons of mass destruction, it was a founding permanent member of the United Nations Security Council as well as a member of the Organization for Security and Co-operation in Europe, the World Federation of Trade Unions and the leading member of the Council for Mutual Economic Assistance and the Warsaw Pact.

The word "Soviet" is derived from a Russian word сове́т meaning council, advice, harmony and all deriving from the proto-Slavic verbal stem of vět-iti, related to Slavic věst, English "wise", the root in "ad-vis-or", or the Dutch weten. The word sovietnik means "councillor". A number of organizations in Russian history were called "council". For example, in the Russian Empire the State Council, which functioned from 1810 to 1917, was referred to as a Council of Ministers after the revolt of 1905. During the Georgian Affair, Vladimir Lenin envisioned an expression of Great Russian ethnic chauvinism by Joseph Stalin and his supporters, calling for these nation-states to join Russia as semi-independent parts of a greater union, which he named as the Union of Soviet Republics of Europe and Asia. Stalin resisted the proposal, but accepted it, although with Lenin's agreement changed the name of the newly proposed sta

Richard Hamming

Richard Wesley Hamming was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code, the Hamming window, Hamming numbers, sphere-packing, the Hamming distance. Born in Chicago, Hamming attended University of Chicago, University of Nebraska and the University of Illinois at Urbana–Champaign, where he wrote his doctoral thesis in mathematics under the supervision of Waldemar Trjitzinsky. In April 1945 he joined the Manhattan Project at the Los Alamos Laboratory, where he programmed the IBM calculating machines that computed the solution to equations provided by the project's physicists, he left to join the Bell Telephone Laboratories in 1946. Over the next fifteen years he was involved in nearly all of the Laboratories' most prominent achievements. After retiring from the Bell Labs in 1976, Hamming took a position at the Naval Postgraduate School in Monterey, where he worked as an adjunct professor and senior lecturer in computer science, devoted himself to teaching and writing books.

He delivered his last lecture in December 1997, just a few weeks before he died from a heart attack on January 7, 1998. Richard Wesley Hamming was born in Chicago, Illinois, on February 11, 1915, the son of Richard J. Hamming, a credit manager, Mabel G. Redfield, he grew up in Chicago, where he attended Crane Junior College. Hamming wanted to study engineering, but money was scarce during the Great Depression, the only scholarship offer he received came from the University of Chicago, which had no engineering school. Instead, he became a science student, majoring in mathematics, received his Bachelor of Science degree in 1937, he considered this a fortunate turn of events. "As an engineer," he said, "I would have been the guy going down manholes instead of having the excitement of frontier research work."He went on to earn a Master of Arts degree from the University of Nebraska in 1939, entered the University of Illinois at Urbana–Champaign, where he wrote his doctoral thesis on Some Problems in the Boundary Value Theory of Linear Differential Equations under the supervision of Waldemar Trjitzinsky.

His thesis was an extension of Trjitzinsky's work in that area. He looked at Green's function and further developed Jacob Tamarkin's methods for obtaining characteristic solutions. While he was a graduate student, he read George Boole's The Laws of Thought; the University of Illinois at Urbana–Champaign awarded Hamming his Doctor of Philosophy in 1942, he became an Instructor in Mathematics there. He married Wanda Little, a fellow student, on September 5, 1942 after she was awarded her own Master of Arts in English literature, they had no children. In 1944, he became an Assistant Professor at the J. B. Speed Scientific School at the University of Louisville in Louisville, Kentucky. With World War II still ongoing, Hamming left Louisville in April 1945 to work on the Manhattan Project at the Los Alamos Laboratory, in Hans Bethe's division, programming the IBM calculating machines that computed the solution to equations provided by the project's physicists, his wife Wanda soon followed, taking a job at Los Alamos as a human computer, working for Bethe and Edward Teller.

Hamming recalled that:Shortly before the first field test, a man asked me to check some arithmetic he had done, I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen—after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, left. I said to myself, "What have you done, you are involved in risking all of life, known in the Universe, you do not know much of an essential part?" I was pacing down the corridor when a friend asked me what was bothering me. I told him, his reply was, "Never mind, Hamming, no one will blame you." Hamming remained at Los Alamos until 1946, when he accepted a post at the Bell Telephone Laboratories.

For the trip to New Jersey, he bought Klaus Fuchs's old car. When he sold it just weeks before Fuchs was unmasked as a spy, the FBI regarded the timing as suspicious enough to interrogate Hamming. Although Hamming described his role at Los Alamos as being that of a "computer janitor", he saw computer simulations of experiments that would have been impossible to perform in a laboratory. "And when I had time to think about it," he recalled, "I realized that it meant that science was going to be changed". At the Bell Labs Hamming shared an office for a time with Claude Shannon; the Mathematical Research Department included John Tukey and Los Alamos veterans Donald Ling and Brockway McMillan. Shannon, Ling, McMillan and Hamming came to call themselves the Young Turks. "We were first-class troublemakers," Hamming recalled. "We still got valuable results. Thus management had to tolerate us and let us alone a lot of the time."Although Hamming had been hired to work on elasticity theory, he still spent much of his time with the calculating machines.

Before h

Abraham Lempel

Abraham Lempel is an Israeli computer scientist and one of the fathers of the LZ family of lossless data compression algorithms. Lempel was born on 10 February 1936 in Poland, he studied at Technion - Israel Institute of Technology, received a B. Sc. in 1963, M. Sc. in 1965, D. Sc. in 1967. Since 1977 he has held the title of full professor. Lempel is now a professor emeritus in Technion, his important works start with the presentation of the LZ77 algorithm in a paper entitled "A Universal Algorithm for Sequential Data Compression" in the IEEE Transactions on Information Theory, co-authored by Jacob Ziv. He is the recipient of the 1998 Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society. Lempel founded HP Labs—Israel in 1994, served as its director until October 2007; the LZ77 and LZ78 algorithms authored by Lempel and Jacob Ziv have led to a number of derivative works, including the Lempel–Ziv–Welch algorithm, used in the GIF image format, the Lempel-Ziv-Markov chain algorithm, used in the 7-Zip and xz compressors.

The algorithms have been used as published in formats such as DEFLATE, used in the PNG image format. Jacob Ziv, Abraham Lempel. "A Universal Algorithm for Sequential Data Compression". IEEE Transactions on Information Theory. 23: 337–343. CiteSeerX 10.1.1.118.8921. Doi:10.1109/TIT.1977.1055714. Timeline of algorithms Data compression Oblivious transfer Abraham Lempel - GHN: IEEE Global History Network Abraham Lempel at DBLP Bibliography Server Technion: Computer Science Department: Prof. Abraham Lempel Abraham Lempel: Senior HP Fellow at the Wayback Machine Abraham Lempel at the Mathematics Genealogy Project

Error correction code

In computing, telecommunication, information theory, coding theory, an error correction code, sometimes error correcting code, is used for controlling errors in data over unreliable or noisy communication channels. The central idea is the sender encodes the message with a redundant in the form of an ECC; the American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming code. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, to correct these errors without retransmission. ECC gives the receiver the ability to correct errors without needing a reverse channel to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. ECC is therefore applied in situations where retransmissions are costly or impossible, such as one-way communication links and when transmitting to multiple receivers in multicast. For example, in the case of a satellite orbiting around Uranus, a retransmission because of decoding errors can create a delay of 5 hours.

ECC information is added to mass storage devices to enable recovery of corrupted data, is used in modems, is used on systems where the primary memory is ECC memory. ECC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, ECC is an integral part of the initial analog-to-digital conversion in the receiver; the Viterbi decoder implements a soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. Many ECC encoders/decoders can generate a bit-error rate signal which can be used as feedback to fine-tune the analog receiving electronics; the maximum fractions of errors or of missing bits that can be corrected is determined by the design of the ECC code, so different error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective signal-to-noise ratio.

The noisy-channel coding theorem of Claude Shannon answers the question of how much bandwidth is left for data communication while using the most efficient code that turns the decoding error probability to zero. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, hence gives no insight of how to build a capacity achieving code. After years of research, some advanced ECC systems nowadays come close to the theoretical maximum. ECC is accomplished by adding redundancy to the transmitted information using an algorithm. A redundant bit may be a complex function of many original information bits; the original information may or may not appear in the encoded output. A simplistic example of ECC is to transmit each data bit 3 times, known as a repetition code. Through a noisy channel, a receiver might see table below; this allows an error in any one of the three samples to be corrected by "majority vote" or "democratic voting".

The correcting ability of this ECC is: Up to 1 bit of triplet in error, or up to 2 bits of triplet omitted. Though simple to implement and used, this triple modular redundancy is a inefficient ECC. Better ECC codes examine the last several dozen, or the last several hundred received bits to determine how to decode the current small handful of bits. ECC could be said to work by "averaging noise"; because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum signal-to-noise ratio and not at all below it. This all-or-nothing tendency – the cliff effect – becomes more pronounced as stronger codes are used that more approach the theoretical Shannon limit. Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits. Most telecommunication systems use a fixed channel code designed to tolerate the expected worst-case bit error rate, fail to work at all if the bit error rate is worse.

However, some systems adapt to the given channel error conditions: some instances of hybrid automatic repeat-request use a fixed ECC method as long as the ECC can handle the error rate switch to ARQ when the error rate gets too high. The two main categories of ECC codes are convolutional codes. Block codes work on fixed-size blocks of symbols of predetermined size. Practical block codes can be hard-decoded in polynomial time to their block length. Convolutional codes work on symbol streams of arbitrary length, they are most soft decoded with the Viterbi algorithm, though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of exponentially increasing complexity. A convolu