J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, Berlin, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some i, e i 2 C t , this implies C t = Z n 2 . So H(n) C t has degree n and edge connectivity n. 2 Cyclic Codes To complete we need to nd some good sequence of code. For this we brieAEy recall some basics of the theory of error correcting codes. The reader will nd more information in [18] or [24]. Cyclic codes are very convenient since they can be viewed as principal ideal of the ring R = F 2 [x] x n Gamma 1) In other words, a cyclic code C consists of all multiples of a polynomial g(x) called generator polynomial. Note that this polynomial must be a factor of x n Gamma 1. The ....

J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, Berlin, 1982.

....strings of binary digits, the symbols having probabilities fl i . Each symbol s code is defined by the sequence of binary digits in tracing the path from the root to each symbol s leaf. The Huffman construction generates a prefix free binary code of the minimum expected length, see for instance [10]. We note that the key step of summing probabilities fl i in the Huffman algorithm corresponds exactly to multiplying the intermediate products of the q i in the binary tree. Conveniently, precisely these products appear in the exponentiation algorithm. The optimization procedure is formalized as ....

van Lint, J.H. "Introduction to Coding Theory", Graduate Texts in Mathematics, vol 86 3rd edition, Springer, March 1999.

....(t) lim inf n 1 maxR(C n ) where the maximum is computed over all t identifying codes of length n. Note that for alphabet sizes q t 2 theorem 1 does not prove that R q (t) 0 (for example because (n; M) codes that satisfy the distance condition must have M d, see Plotkin s bound e.g. in [11]) In fact, non trivial t identifying codes do not always exist if the alphabet size q is not big enough. For example, Boneh and Shaw [3] note that if q = 2 then no 2 identifying code C exists with jCj 3. To see this suppose that x; y; z are distinct codewords of C: define s by choosing s i to ....

J. H. van Lint, Introduction to Coding Theory, Grad. Texts in Math. 86, Springer-Verlag 1982.

....quickly can a SRW from a random uniformly chosen point of A reach a distribution that is almost uniform on n A reasonable guess is that the best choice would be to take A itself to be random. This is closely related to the conjecture that the Gilbert Varshamov bounds for codes are optimal; see [142]. Cliques in graphs This time, let the variables correspond to the n = m 2 edges of the complete graph with m vertices. Every assignment of values to the variables corresponds to a graph: the graph whose edges correspond to the variables with value 1. Let f be the size of the largest ....

J. H. van Lint, Introduction to coding theory. Third edition. Graduate Texts in Mathematics, 86. Springer-Verlag, Berlin, 1999. 53

....called code words. The minimum distance of C is the least Hamming distance between two distinct code words. One of the main open problems in coding theory is to determine the largest cardinality, A(n; d) of a binary code of length n and minimal distance d. For more information on coding theory see [13, 15, 14]. Our main concern is with the case where d is proportional to n. When n tends to infinity and d=n tends to ffi 1=2, then A(n; d) is exponential in n. The determination of the basis for this exponential function is a difficult question of fundamental importance for coding theory. We need some ....

....observed in [12] fi(x) is a nonnegative combination of Krawtchouk polynomials. Now, apply the previous Proposition for m = sn. With this choice t = ff(s) o(1) n. As before, it suffices to consider the largest term on right hand side of (13) which we proceed to do. As shown in [12] see also [15], p.67) fi 0 = Gamma 2 t 1 Gamma n t Delta K t (a)K t 1 (a) Therefore, fi(0) Delta fi Gamma1 0 = n 1) 2 2a(t 1) Gamma n t Delta and n Gamma1 log(fi(0) Delta fi Gamma1 0 ) H(t) o(1) H(ff(s) o(1) Denote i = x Delta n and calculate the i th term in the sum: n ....

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J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, 1982.

....some i, e i 2 C t , this implies C t = Z n 2 . So H(n) C t has degree n and edge connectivity n. 2 Cyclic Codes To complete we need to nd some good sequence of code. For this we brieAEy recall some basics of the theory of error correcting codes. The reader will nd more information in [18] or [24]. Cyclic codes are very convenient since they can be viewed as principal ideal of the ring R = F 2 [x] x n Gamma 1) In other words, a cyclic code C consists of all multiples of a polynomial g(x) called generator polynomial. Note that this polynomial must be a factor of x n Gamma 1. The ....

J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, Berlin, 1982.

....The greater part of the discussion of the solved instances of SS in Section 1.2.2 was taken from [IN96] A proof of Section 1.3.2 s Lemma 17 can be found in [TV91, p. 77] It can be proven similarly to the Gilbert Varshamov Bound [PW72, Section 4. 1] An alternate discussion can be found in [vL82, Section 5.1] More historical details about Shannon s Theorems can be found in [Abr63, p. 149, 174] Other discussions and proofs of Shannon s Theorems (for not necessarily linear codes) can be found in [vL82, Chapter 2] PW72, Section 4.2] and [Ham86, Chapter 10] The Converse of Shannon s ....

....Gilbert Varshamov Bound [PW72, Section 4.1] An alternate discussion can be found in [vL82, Section 5.1] More historical details about Shannon s Theorems can be found in [Abr63, p. 149, 174] Other discussions and proofs of Shannon s Theorems (for not necessarily linear codes) can be found in [vL82, Chapter 2] PW72, Section 4.2] and [Ham86, Chapter 10] The Converse of Shannon s Theorem can be found in [SW64, p. 71] Ham86, Section 10.8] and [Abr63, Chapter 6] Other coding theory references include [MS77] Gal68] and [Sti95, Chapter 10] The facts concerning approximation schemes ....

J. H. van Lint. Introduction to Coding Theory. Springer-Verlag, New York, Heidelberg, Berlin, 1982.

....to the cardinality jAj. 3.7) Definition. Entropy Function. For 0 x 1=2 let H(x) x log 2 1 x (1 Gamma x) log 2 1 1 Gamma x : We agree that H(0) 0. Thus H is an increasing concave function on the interval [0; 1=2] We use the following estimate (see, for example, Theorem 1.4. 5 of [van Lint 99] 3:7:1) r X k=0 n k 2 nH(r=n) for r n=2: Also, we remark that around x = 0 we have (3:7:2) H(x) x log 2 1 x O(x) and H i 1 2 Gamma x j = 1 Gamma 2 ln 2 x 2 O(x 3 ) We will use the classical isoperimetric inequality for the Boolean cube (see, for example, ....

J.H. van Lint, Introduction to Coding Theory, Third edition. 33 Graduate Texts in Mathematics, 86, Springer-Verlag, Berlin, 1999.

....Netherlands. This research was partially supported by the Netherlands organization for scientific research (NWO) 1 Although this paper is quite self contained, a certain knowledge of algebraic geometry is taken for granted. For this, we refer to [2] 4] 11] 16] or [22] For coding theory, see [15], 16] or [17] Outline of the paper In Section II we define weakly algebraic geometric (WAG) algebraic geometric (AG) and strongly algebraic geometric (SAG) codes (Definition 2) The class of SAG codes is a proper subset of the class of AG codes, and the class of AG codes is a proper subset of ....

J.H. van Lint, Introduction to coding theory. Graduate Texts in Math. 86. SpringerVerlag, Berlin Heidelberg New York, 1982. 41

....tone of the lectures, and I hope that the reader will nd this monograph both accessible and useful. Exercises are scattered throughout, and the reader is strongly encouraged to work through them. Of the sources listed in the bibliography, it should be pointed out that [CLO2] Ga] H] [L], MS] NZM] and [S] were used most intensively in preparing these notes. In particular: Theorem 1.12, which gives some important properties of cyclic codes, can be found in [MS] The proof given for the Singleton Bound (Theorem 2.1) is from [S] v vi Preface The proofs given for the ....

....important properties of cyclic codes, can be found in [MS] The proof given for the Singleton Bound (Theorem 2.1) is from [S] v vi Preface The proofs given for the Plotkin Bound (Theorem 2.3) the Gilbert Varshamov Bound (Theorem 2.4) and the asymptotic Plotkin Bound (Theorem 2. 7) are from [L]. Exercise 3.6, about nding points on a hyperbola, is taken from [NZM] The pictures and examples of singularities (as in Exercise 4.4) are from [H] The proof of the classi cation of nite elds outlined in the Exercises in Section B.3 is from [CLO2] More generally, the reader is ....

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J. H. Van Lint, Introduction to Coding Theory, Third Edition. Springer-Verlag, New York, 1999.

....and Lovasz [Lov90] and the forthcoming bookby Kushilevitz and Nisan [KN95] for excellent introductions to communication complexity. 2 Protocols based on the Hamming error correcting code The protocols described in this section are based on the Hamming error correcting code (see van Lint [vL91]) Similar coding ideas were employed by Lupanov [Lup73] and Gaskov [Gas78] For every x; y 2 f0; 1g n , we let d(x; y) be the Hamming distance between x and y, i.e. the number of positions in which x and y differ. For every x 2 f0; 1g n , we let ball(x) fy 2 f0; 1g n j d(x; y) 1g ; ....

J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, 1991. Second Edition. 13

....some conclusions and open problems. This survey draws heavily on references [5] and [6] which contain full details and proofs, as well as an account of prior and independent work on power control in OFDM using Golay complementary sequences. For further background on classical coding theory, see [7] or [8] 2 OFDM Transmission We begin by describing the signals in an OFDM system and introducing some associated terminology. An n carrier OFDM signal is composed by adding together n equally spaced, phase shifted sinusoidal carriers. Information is carried in the phase shift applied to each ....

....instead 8 of the 12 Golay cosets , we obtain a code still having a PMEPR of 2, but with an increased rate of 0.50 and decreased minimum Hamming distance of 4. A compromise option can be obtained using four out of the six cosets identified by Corollary 6. 1 that lie in the Kerdock code of length 16 [7]. These six cosets have representatives: x 1 x 2 x 2 x 4 x 3 x 4 ; x 1 x 3 x 2 x 3 x 2 x 4 ; x 1 x 4 x 3 x 4 x 2 x 3 ; x 1 x 2 x 1 x 3 x 3 x 4 ; x 2 x 4 x 1 x 4 x 1 x 3 ; x 2 x 3 x 1 x 2 x 1 x 4 : The resulting code has rate 0.44 and minimum Hamming distance 6. A ....

J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, Berlin, 2nd edition, 1992.

....results of 50 runs are shown as a function of generation index. on every problem resulted in a different solution that are considerably far from each other (see Table 1) The many optimal solutions found do not seem to have any common feature except the bit distribution. Results of coding theory [Lint 1992] also support that a great number of paths can exist without interfering with each other. As it was shown in [Jelasity et al. 1995] GAS, a GA with a special niching technique supporting the separate handling of different local optima outperformed the standard GA on this problem. Finally, the ....

J.H. van Lint (1992) Introduction to Coding Theory., Springer-Verlag.

....solve, say in NC 1 . It turns out that this is precisely what we do, but we are faced with numerous technical obstacles, which we solve in the course of the proof. Proof. It is known that the polynomial X 2 Delta3 X 3 1 2 Z 2 [X] is an irreducible polynomial over Z 2 for all 0 [vL91] In the following, by a finite field GF(2 m ) where m= 2 Delta 3 , we refer explicitly to the field Z 2 [X] X 2 Delta3 X 3 1) Let S be a sparse set hard for P under many one reductions. With a view to creating a Vandermonde system of equations, we will consider the ....

....relies on solving a system of equations where a fraction of the equations could be erroneous. We then derandomize this algorithm, taking advantage of certain error correcting capabilities of the set D used before for the many one case. The set D is formally identical to an error correcting code [vL91] that is obtained by concatenating a certain Reed Solomon code with a Hadamard code. It is an indication of the effectiveness of algebraic and derandomization techniques that the proof for the many one case can be generalized to account for the case of bounded truth table reductions. We note ....

J.H. van Lint. Introduction to Coding Theory. Springer-Verlag, 1991.

....In this lecture, we will first formalize the notion of error correcting codes and list the various parameters of ECCs that have been considered in the literature. We will then discuss some particular error correcting codes, namely, the Reed Solomon code, the Reed Muller code and the Hadamard code [Lin82]. We will also discuss Forney s method of composing any two codes to get a new error correcting code. 1.1 PCPs and Error Correcting Codes According to the PCP theorem [BLFS91, Has97] for any language L 2 NP , we have: 8x 2 L; 9 proof Pi s:t: E R [V Pi (x; R) 1 Gamma ffl Now, consider ....

J H Van Lint. Introduction to Coding Theory. Springer Verlag, 1982. 10-6

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J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York, 1982.

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Van Lint, J.H., Introduction to Coding Theory, 2nd Edition, Springer--Verlag, 1992.

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J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York (1982).

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J. H. van Lint, Introduction to Coding Theory (Springer, New York, 1982).

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J. H. van Lint, Introduction to Coding Theory (Springer, New York, 1982).

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J.H. van Lint, Introduction to Coding Theory, Second Addition, Springer-Verlag, 1991.

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J. H. van Lint. Introduction to Coding Theory. Springer-Verlag, 1982.

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J. H. Van Lint. Introduction to Coding Theory. Springer-Verlag, 1992. 12

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J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York, 1982.

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J.H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York, 1982.

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