J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18: 652--656, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....define the (log m 1) qubit state as i (x)# for each x , where e i (x) is the i th bit of e (x) This is called a quantum fingerprint of x. Since two distinct codewords can be equal in at most #m positions, for any x y we have #h # # #m m = #. Justesen codes [7] is a reasonable choice of such codes, which give # 9 10 1 (15c) for any chosen c 2. Distinguishing can be done with one sided error probability by the procedure that measures and outputs the first qubit of the state (H# I) controlled SWAP ) H# I) 0# h x where H is the ....

Justesen, J. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory 18 (1972), 652--656.

....of Ramsey graphs is based on combining many graphs, most of which are known to be good, without testing which are the bad ones. The fact that most of graphs are good compensates for the bad ones. Remark: Our construction is inspired by Justesen s construction of good error correcting codes [9] (see [10] for more information on codes) It was known that good binary codes can be obtained by concatenating a good code over a large alphabet (which can be explicitly constructed) with good codes that map the large alphabet to the small one. The existence of this code was only proved ....

J. Justesen, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory, IT-18 (1972), 652--656.

....the best inner code [9] This construction is sometimes not explicit enough , as the construction complexity is not a fixed polynomial independent of ffi. The first explicit asymptotically good construction of very large minimum distance was given by Weldon [32] following the work of Justesen [17], and related constructions appear in the [29, 30] One can also use algebraic geometric codes as outer codes in a concatenation scheme to give explicit construction of codes with n = O(k=ffi 3 ) but these codes inherit the high construction complexity of algebraic geometric codes. The best ....

....n = poly(k=fl) can be efficiently list decoded up to a fraction (1 Gamma 1=q Gamma fl) of errors [28] this is also implicit in [19] but even for this code the dependence of n on fl was not optimized. The situation is worse for asymptotically good codes. In Justesen s original paper [17], he also gives an algorithm to decode his code construction up to half the minimum distance. Note that unambiguous decoding (as opposed to list decoding) implies that we cannot hope to recover from more than 1 2 (1 Gamma 1=q) fraction of errors as any q ary code (with exponentially many ....

[Article contains additional citation context not shown here]

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18 (1972), pp. 652-656.

....codewords E(x) and E(y) with x 6= y) is at least (1 Gamma ffi )m, where c and ffi are constants. For the particular case of Justesen codes, we may choose any c 2 and we will have ffi 9=10 1= 15c) assuming n is sufficiently large) For further information on Justesen codes, see Justesen [Jus72] and MacWilliams and Sloane [MS77, Chapter 10] Now, for x 2 f0; 1g n and i 2 f1; 2; mg, let E i (x) denote the i th bit of E(x) The shared key is a random i 2 f1; 2; mg (which consists of log(m) 2 log(n) O(1) bits) Alice and Bob respectively send the bits E i (x) and E i (y) ....

J. Justesen, A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18:652--656, 1972.

....=2 of errors in linear time. Since = 1 ) 1=2 ) and 0 is arbitrary, this will imply the claimed result. To this end, we use the following general result on decoding concatenated codes. This result appears to be folklore and has been observed and used by several authors, including Justesen [6] who gave an algorithm to decode his asymptotically good concatenated code construction up to the product bound (i.e. half the product of the distances of the outer and inner codes) The basic idea and inspiration behind this method comes from the work of Forney [4] on Generalized Minimum ....

....by the nal algorithm A is at most d (and hence the dT out term in the runtime) and independent of the outer blocklength N . This will be crucial for us. The fact that GMD decoding allows for such a few invocations of the outer decoding algorithm has been observed before (for example, by Justesen [6] who makes a one line comment on it) A formal proof of the above proposition can be found, for example, in [5, Appendix A] Since the relative distance of the inner code C 3 in our concatenated construction C is at least (1=2 ) it can be uniquely decoded up to a fraction (1=2 ) 2 of ....

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory, 18:652-656, 1972.

....type. The code can be encoded and decoded in polynomial time up to its designed distance. Reversals of segments of the codeword can also be accomodated using the methods described. This is a generalization of the constructive, asymptotically good codes for the Hamming distance given by Justesen [9]. Those codes could correct only alterations of characters, whereas here we allow more general errors. Channels with insertions and deletions occur in various situations, for example: ffl Insertion and deletion errors occur in reading magnetic and optical media (in addition to the more familiar ....

....this class (subject to limits on the error probabilities) We emphasize that our codes allow arbitrary insertions, deletions and transpositions, subject only to numerical limits; the errors do not have to be of restricted types, or distributed randomly. 2 The Code Our code, like a Justesen code [9], is a two level code. The outer level can be given by polynomial evaluation, i.e. a Reed Solomon code [14] and decoded using the Welch Berlekamp algorithm [22] see also [8, 3] or by any asymptotically good, efficiently encodable and decodable code. The inner level is given by a code which we ....

J. Justesen, "A Class of Constructive, Asymptotically Good Algebraic Codes," IEEE Trans. Inform. Theory, September 1972, 18:652-656.

....eld. Wozencraft shows that one of the codes C also achieves the same performance as guaranteed by the results of Gilbert and Varshamov. Wozencraft s ensemble is actually a wider class of constructions, of which the above is one speci c example. This is the example given explicitly in Justesen [51]. To understand the bound (q k V q (n; d) q n ) in slightly clearer light, let us simplify it slightly. For 0 1, let H q ( log q q 1 (1 ) log q 1 1 : Then V q (n; n) is well approximated by q Hq ( n . The results of Gilbert, Varshamov (and Wozencraft) show ....

Jrn Justesen. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory, 18:652-656, 1972. 16

....E : f0; 1g m 7 f0; 1g q for mapping words in f0; 1g m to C. What are the requirement from the code As we shall see, q Delta log 2 2 Gammaffl must be at least 3n, and we want q=m to be a fixed constant. Such codes exist, and specifically the Justesen code is a constructive example [Ju] . For the amortization to work it sufficient that m be linear in n. For a vector R = r 1 ; r 2 ; r k ) with r i 2 f0; 1g and with exactly q indices i such that r i = 1 let G R (s) denote the vector A = a 1 ; a 2 ; a q ) where a i = B j(i) s) and j(i) is the index of the ....

J. Justesen, A class of constructive asymptotically good algebraic codes , IEEE Transactions on Information Theory 18 (1972), pp. 652-656.

....any two distinct codewords E(x) and E(y) with x 6= y) is at least (1 )m, where c and are constants. For the particular case of Justesen codes, we may choose any c 2 and we will have 9=10 1= 15c) assuming n is suciently large) For further information on Justesen codes, see Justesen [Jus72] and MacWilliams and Sloane [MS77, Chapter 10] Now, for x 2 f0; 1g n and i 2 f1; 2; mg, let E i (x) denote the i th bit of E(x) The shared key is a random i 2 f1; 2; mg (which consists of log(m) 2 log(n) O(1) bits) Alice and Bob respectively send the bits E i (x) and E i (y) ....

J. Justesen, A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18:652-656, 1972.

....for the best inner code [9] This construction is sometimes not explicit enough , as the construction complexity is not a xed polynomial independent of . The rst explicit asymptotically good construction of very large minimum distance was given by Weldon [32] following the work of Justesen [17], and related constructions appear in the [29, 30] One can also use algebraic geometric codes as outer codes in a concatenation scheme to give explicit construction of codes with n = O(k= 3 ) but these codes inherit the high construction complexity of algebraic geometric codes. The best code ....

....Hadamard code with blocklength n = poly(k= can be eciently list decoded up to a fraction (1 1=q ) of errors [28] this is also implicit in [19] but even for this code the dependence of n on was not optimized. The situation is worse for asymptotically good codes. In Justesen s original paper [17], he also gives an algorithm to decode his code construction up to half the minimum distance. Note that unambiguous decoding (as opposed to list decoding) implies that we cannot hope to recover from more than 1 2 (1 1=q) fraction of errors as any q ary code (with exponentially many codewords) ....

[Article contains additional citation context not shown here]

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18 (1972), pp. 652-656.

....of nondeterministic moves) and running time than all of the previous ones. The idea of using error correcting codes is mentioned by Naor and Naor [NN93] and ascribed to Bruck, referring the reader to [ABN 92] for details. However, the construction in [ABN 92] uses a code of Justesen (see [Jus72, MS77] whose implementation in our setting seems to require exponentiation of field elements of length polynomial in n, which is not known to be computable in (randomized) quasilinear time (cf. AMV88, Sti90] Our point is that by scaling down the size of the field used for basic arithmetic, and ....

....have rate R = K=N = 2 q =2 (2q Gamma: which tends to 0 as q increases. Families of codes are known for which R (as well as ffi) stays bounded below by a constant; such (families of) codes are called good . Good codes require only q O(1) random bits in the above construction. The codes in [Jus72, TV91, ABN 92, JLJH92, She93] are good. However, we do not know of any good codes which give quasi linear runtime in the above construction. 4. Search Versus Decision in Quasilinear Time The classical method of computing partial, multivalued functions using sets as oracles is the prefix set ....

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Info. Thy., IT-18:652--656, September 1972.

....given in section 3.3. 3.1 Error corrected random sources In this subsection we show that if we apply an error correcting code to an arbitrary k source, we obtain a k source which has k indices which induce an k) source. We use the following construction of error correcting codes. Theorem 4 [Jus72] There exist constants a; b and an explicit error correcting code EC : f0; 1g n f0; 1g an such that for every x 1 6= x 2 2 f0; 1g n , d(EC(x 1 ) EC(x 2 ) bn. d(z 1 ; z 2 ) denotes the Hamming distance between z 1 ; z 2 ) In the remainder of this section we x a; b and EC to be these ....

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Info. Theory, 18:652-656, 1972.

....enough, as the construction complexity is not a fixed polynomial independent of , and also the entries of the generator matrix cannot be computed in polylogarithmic (in n) time. The first explicit asymptotically good construction achieving this was given by [23] following the work of Justesen [10], and related constructions appear in [20, 21] Constructions of algebraic geometric codes imply such codes with n = O(k= 3 ) but these codes have very high construction complexity. The best constructions with reasonable complexity is due to [3] which achieves n = O(k= 3 ) For the case of ....

....codes with Hadamard codes can achieve n = poly(k= to decode up to a fraction (1 1=q ) of errors [19] this is also implicit in [12] but even for this code the dependence of n on was not optimized. The situation is worse for asymptotically good codes. In Justesen s original paper [10], he also gives an algorithm to decode his code construction to half the minimum distance. Note that unique codeword decoding immediately implies that we cannot hope to recover from more than 1 2 (1 1=q) fraction of errors as any q ary code (with exponentially many codewords) has distance less ....

[Article contains additional citation context not shown here]

J. JUSTESEN. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18 (1972), pp. 652-656.

.... enough, as the construction complexity is not a fixed polynomial independent of ffi, and also the entries of the generator matrix cannot be computed in polylogarithmic (in n) time. The first explicit asymptotically good construction achieving this was given by [23] following the work of Justesen [10], and related constructions appear in [20, 21] Constructions of algebraic geometric codes imply such codes with n = O(k=ffi 3 ) but these codes have very high construction complexity. The best constructions with reasonable complexity is due to [3] which achieves n = O(k=ffi 3 ) For the ....

....of Reed Solomon codes, Hadamard codes or concatenation, can find a brief description in Section 2. rors [19] this is also implicit in [12] but even for this code the dependence of n on fl was not optimized. The situation is worse for asymptotically good codes. In Justesen s original paper [10], he also gives an algorithm to decode his code construction to half the minimum distance. Note that unique codeword decoding immediately implies that we cannot hope to recover from more than 1 2 (1 Gamma 1=q) fraction of errors as any q ary code (with exponentially many codewords) has distance ....

[Article contains additional citation context not shown here]

J. JUSTESEN. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18 (1972), pp. 652-656.

....: jJ j c sec Delta mg, and ff 2 f0; 1g jIj , Prob(E(x) I = ff) Prob(E(y) I = ff) Furthermore, E(x) I is uniformly distributed over f0; 1g jIj . In addition, on input x, algorithm E uses O(jxj) coin tosses. Items 1 and 2 are standard requirements of Coding Theory, firstly met by Justesen [6]. What is non standard in the above is Item 3. Indeed, Item 3 is impossible if one insists that the encoding algorithm (i.e. E) is deterministic. 2.1 Proof of Theorem 1 Using a nice error correcting code, the key idea is to encode the information by first augmenting it by a sufficiently long ....

....rank, and therefore y Delta B is uniformly distributed (and so is z Delta M 0 regardless of x) What is missing in the above is a specific construction satisfying the hypothesis as well as allowing efficient decoding. Such a construction can be obtained by mimicking Justesen s construction [6]. Recall that Justesen s Code is obtained by composing two codes: Specifically, an outer linear code over an n symbol alphabet is composed with an inner random linear code. 1 The outer code is obtained by viewing the message as the coefficients of a polynomial of degree t Gamma 1 over a field ....

[Article contains additional citation context not shown here]

J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18: 652--656, 1972.

.... bound [2] Moreover, the same is valid even if A is a fixed MDS (say Reed Solomon) code [3] 9] By making n grow slower than N and taking varying inner and MDS outer codes, it is possible to present families of codes with both nonvanishing rate and distance and low construction complexity [11] [5]. Improvements of the parameters of these families are based on multilevel concatenations [3] and the use of algebraic geometry codes [6] A detailed account of the early work is given in [3] see also overviews in [4] 1] For a given rate R 2 [0; 1] denote by ffi 0 = ffi 0 (R) the GV distance, ....

J. Justesen, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory 18 (1972), 652--656.

.... bound [2] Moreover, the same is valid even if A is a fixed MDS (say Reed Solomon) code [3] 9] By making n grow slower than N and taking varying inner and MDS outer codes, it is possible to present families of codes with both nonvanishing rate and distance and low construction complexity [11] [5]. Improvements of the parameters of these families are based on multilevel concatenations [3] and the use of algebraic geometry codes [6] A detailed account of the early work is given in [3] see also overviews in [4] 1] For a given rate R 2 [0; 1] denote by ffi 0 = ffi 0 (R) the GV distance, ....

J. Justesen, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory 18 (1972), 652--656.

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J. Justesen. A class of constructive asymptotically good algebraic codes. IEEE Trans. Inform. Theory, 18: 652--656, 1972.

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J. Justesen, "A class of constructive asymptotically good algebraic codes," IEEE Trans. Inform. Theory, vol. IT-18, pp. 652--656, 1972.

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Jrn Justesen. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory, 18:652-656, 1972.

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J. Justesen, `A class of constructive asymptotically good algebraic codes', IEEE Trans. Inform. Theory, vol. 18, pp. 652--656, Sept. 1972.

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Jrn Justesen. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory, 18:652--656, 1972.

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J. Justesen, "A Class of Constructive, Asymptotically Good Algebraic Codes," IEEE Trans. Inform. Theory, September 1972, 18:652-656.

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Justesen, J. A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory 18 (1972), 652--656.

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J. Justesen, A Class of Constructive Asymptotically Good Algebraic Codes, IEEE Transactions on Information Theory, 18 pp. 652-656, 1972.

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