By Dave K. Kythe

ISBN-10: 1439881812

ISBN-13: 9781439881811

Using an easy but rigorous process, Algebraic and Stochastic Coding concept makes the topic of coding thought effortless to appreciate for readers with a radical wisdom of electronic mathematics, Boolean and smooth algebra, and likelihood conception. It explains the underlying ideas of coding conception and provides a transparent, special description of every code. extra complicated readers will relish its insurance of contemporary advancements in coding concept and stochastic processes.

After a quick assessment of coding background and Boolean algebra, the publication introduces linear codes, together with Hamming and Golay codes. It then examines codes in accordance with the Galois box idea in addition to their software in BCH and particularly the Reed–Solomon codes which have been used for mistakes correction of information transmissions in house missions.

The significant outlook in coding conception seems aimed toward stochastic procedures, and this ebook takes a daring step during this path. As examine makes a speciality of mistakes correction and restoration of erasures, the publication discusses trust propagation and distributions. It examines the low-density parity-check and erasure codes that experience unfolded new methods to enhance wide-area community information transmission. It additionally describes sleek codes, equivalent to the Luby remodel and Raptor codes, which are permitting new instructions in high-speed transmission of very huge info to a number of users.

This powerful, self-contained textual content absolutely explains coding difficulties, illustrating them with greater than 2 hundred examples. Combining idea and computational suggestions, it is going to charm not just to scholars but additionally to execs, researchers, and teachers in components equivalent to coding idea and sign and picture processing.

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Additional resources for Algebraic and stochastic coding theory

Example text

In computung the decimal (b = 10) and the binary (b = 2) number systems we need a representation or coding of the decimal digits in terms of binary symbols (called bits). This requires at least four bits, and any 10 out of the 16 possible permutations of these four bits represent the decimal digits. 1, where d denotes the decimal digit. In the BCD code, the weights of the positions are the same as in the binary number system, so that each decimal digit is assigned a combination of bits, which is the same as the number represented by the four components regarded as the base 2 number.

An , given a1 ⊕ a2 ⊕ · · · ⊕ an and any (n − 1) of the values, the missing value can be easily determined. Property 3. A string s of bits is called a symbol. A very useful formula is s⊕s=0 for any symbol s. 3 Applications. Some applications involving the above bitwise operations are as follows: The bitwise and operator is sometimes used to perform a bit mask operation, which is used either to isolate part of a string of bits or to determine whether a particular bit is 1 or 0. For example, let the given bit pattern be 0011; then, to determine if the third bit is 1, a bitwise and operation is performed on this bit pattern and another bit pattern containing 1 in the third 22 2 DIGITAL ARITHMETIC bit.

If two bits are different, then the result is 1; but if they are the same, then the result is 0. Thus, for example, 0101 ⊕ 0011 = 0110. In general, if x, y, z are any items, then (i) x ⊕ x = 0, (ii) x ⊕ 0 = x, (iii) x ⊕ y = y ⊕ x, and (iv) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z). In C programming languages, the bitwise xor is denoted by ⊕. 2 Arithmetic and Bitwise Operations Bitwise Operations Arithmetic and Bitwise Operations p q and or xor p q + and or xor 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 10 0 0 0 1 0 0 0 1 0 1 1 0 The bitwise xor operation is the same as addition mod 2.

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