## Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs (1992)

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Venue: | IEEE Transactions on Information Theory |

Citations: | 115 - 22 self |

### BibTeX

@ARTICLE{Alon92constructionof,

author = {Noga Alon and Jehoshua Bruck and Joseph Naor and Moni Naor and Ron M. Roth},

title = {Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs},

journal = {IEEE Transactions on Information Theory},

year = {1992},

volume = {38},

pages = {509--516}

}

### Years of Citing Articles

### OpenURL

### Abstract

A new technique, based on the pseudo-random properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF (2)) as well. Although these concatenated codes lie below Zyablov bound, they are still superior to previously-known explicit constructions in the zero-rate neighborhood.

### Citations

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Citation Context ...sparse random graphs. Expanders, which are the subject of extensive literature, are, roughly, graphs in which every set of at most half of the vertices has many neighbors outside the set. As shown in =-=[2]-=-, the expanding properties of a graph are closely related to the eigenvalues of its adjacency matrix. Since the property we need here is proved by using the eigenvalues, we do not mention the common d... |

299 | Combinatorial Identities - Riordan - 1968 |

259 | Small-bias probability spaces: efficient constructions and applications
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Citation Context ...ght (measured over Φ ∆ ) is “close” to n. The notion of code amplification through expanders has been inspired by recent applications of expanders to deterministic simulation of randomized algorithms =-=[1, 3, 6, 8, 11, 17]-=-. In a way, the application of expanders presented in this paper can be viewed as an improvement on the method introduced in [17], in the sense that the codes that may be obtained are better. These ex... |

218 |
Explicit group-theoretic constructions for combinatorial designs with applications expanders and concentrators
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Citation Context ...∆ − 1 � 1 − O(1/k) � . Therefore, in any infinite family of ∆-regular graphs {Gi = (Vi, Ei)}|Vi|→∞, lim sup λ(Gi) ≥ 2 i→∞ √ ∆ − 1 . (6) Lubotzky, Phillips and Sarnak [14], and independently, Margulis =-=[16]-=-, gave, for every ∆ = p + 1 where p is a prime congruent to 1 modulo 4, explicit constructions of infinite families of ∆-regular graphs Gi with second largest eigenvalues λ(Gi) ≤ 2 √ ∆ − 1. For the sa... |

183 | How to recycle random bits
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Citation Context ...ght (measured over Φ ∆ ) is “close” to n. The notion of code amplification through expanders has been inspired by recent applications of expanders to deterministic simulation of randomized algorithms =-=[1, 3, 6, 8, 11, 17]-=-. In a way, the application of expanders presented in this paper can be viewed as an improvement on the method introduced in [17], in the sense that the codes that may be obtained are better. These ex... |

94 | Dispersers, deterministic amplification, and weak random sources
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Citation Context ...ght (measured over Φ ∆ ) is “close” to n. The notion of code amplification through expanders has been inspired by recent applications of expanders to deterministic simulation of randomized algorithms =-=[1, 3, 6, 8, 11, 17]-=-. In a way, the application of expanders presented in this paper can be viewed as an improvement on the method introduced in [17], in the sense that the codes that may be obtained are better. These ex... |

72 |
Explicit expanders and the Ramanujan conjectures
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Citation Context ...with 6sdiameter k is at least 2 √ ∆ − 1 � 1 − O(1/k) � . Therefore, in any infinite family of ∆-regular graphs {Gi = (Vi, Ei)}|Vi|→∞, lim sup λ(Gi) ≥ 2 i→∞ √ ∆ − 1 . (6) Lubotzky, Phillips and Sarnak =-=[14]-=-, and independently, Margulis [16], gave, for every ∆ = p + 1 where p is a prime congruent to 1 modulo 4, explicit constructions of infinite families of ∆-regular graphs Gi with second largest eigenva... |

69 |
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Citation Context ...stance δ will require the order of nc(δ) operations, where lim 1 δ→1− c(δ) = ∞. Hence, such a code sequence family is q non-uniformly constructive. The exponential search is avoided in Justesen codes =-=[9]-=- and in constructions derived thereof [22, 23, 24, 27], where the inner codes exhaust all members of Wozencraft’s ensemble of randomly shifted codes. Justesen’s construction is also “explicit” in the ... |

46 |
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Citation Context ...s the proof. tb 2 ∆ 2 = � v∈T ≤ � v∈V � |NB(v)| − b∆ � 2 � |NB(v)| − b∆ � 2 ≤ λ 2 b(1 − b)n . In view of the last two results it is natural to ask how far from ∆ the value of λ(G) can be. It is known =-=[2, 18]-=- that the second largest eigenvalue of any ∆-regular graph with 6sdiameter k is at least 2 √ ∆ − 1 � 1 − O(1/k) � . Therefore, in any infinite family of ∆-regular graphs {Gi = (Vi, Ei)}|Vi|→∞, lim sup... |

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Citation Context |

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Citation Context ...g a technique introduced in [17], we construct such a set of size c t 2 3t log m for any fixed t and for sufficiently large m, where c is an absolute constant (independent of t). For related work see =-=[4, 12, 21]-=-. 2. Pseudo-random graphs Expanders are graphs which behave in many ways like sparse random graphs. Expanders, which are the subject of extensive literature, are, roughly, graphs in which every set of... |

36 |
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Citation Context |

36 |
Vector sets for exhaustive testings of logic circuits
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Citation Context ...g a technique introduced in [17], we construct such a set of size c t 2 3t log m for any fixed t and for sufficiently large m, where c is an absolute constant (independent of t). For related work see =-=[4, 12, 21]-=-. 2. Pseudo-random graphs Expanders are graphs which behave in many ways like sparse random graphs. Expanders, which are the subject of extensive literature, are, roughly, graphs in which every set of... |

30 | Multiplicative Number Theory, Second Edition, revised by H.L - Davenport - 1980 |

25 |
Explicit construction of exponential sized families of k-independent sets
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Citation Context ...g a technique introduced in [17], we construct such a set of size c t 2 3t log m for any fixed t and for sufficiently large m, where c is an absolute constant (independent of t). For related work see =-=[4, 12, 21]-=-. 2. Pseudo-random graphs Expanders are graphs which behave in many ways like sparse random graphs. Expanders, which are the subject of extensive literature, are, roughly, graphs in which every set of... |

25 |
Vladut,S.G.,”Modular curves and codes with a polynomial construction
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(Show Context)
Citation Context ...ce δ and of rate R ≥ RGV(δ), where q RGV(δ) ∆ = 1 − Hq(δ) , (1) and Hq(x) ∆ = −x · logq x − (1 − x) · logq(1 − x) + x · logq(q − 1), 0 ≤ x ≤ 1 − 1. Furthermore, q the seminal works of Tsfasman et al. =-=[10, 13, 25]-=- show the existence of good code sequences beyond the Gilbert-Varshamov bound for q ≥ 46. A code sequence S = {Ci} ∞ i=1 over an alphabet Σ is called constructive if there exists an algorithm that com... |

25 |
Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound
- Zink
- 1982
(Show Context)
Citation Context ...ce δ and of rate R ≥ RGV(δ), where q RGV(δ) ∆ = 1 − Hq(δ) , (1) and Hq(x) ∆ = −x · logq x − (1 − x) · logq(1 − x) + x · logq(q − 1), 0 ≤ x ≤ 1 − 1. Furthermore, q the seminal works of Tsfasman et al. =-=[10, 13, 25]-=- show the existence of good code sequences beyond the Gilbert-Varshamov bound for q ≥ 46. A code sequence S = {Ci} ∞ i=1 over an alphabet Σ is called constructive if there exists an algorithm that com... |

22 |
A time randomness tradeoff
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Citation Context |

21 |
An estimate of the complexity of constructing binary linear concatenated codes
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Citation Context ...nd a code which attains the Gilbert-Varshamov bound as the inner code, one can obtain a family of constructive sequences whose rate function R(δ) satisfies the Zyablov bound 1 ni ,sR(δ) ≥ RZyablov(δ) =-=[29]-=-, where RZyablov(δ) ∆ = max δ≤µ≤1− 1 � � � 1 − Hq(µ) 1 − q δ � µ . (2) However, searching for the inner code by any known algorithm requires time complexity which is exponential in the inner code leng... |

10 |
Private communication
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Citation Context ... fixed π, the family {G(p, π)}p is an optimal set of pseudo-random graphs as it attains the bound (6). Although the construction given in [14] and [16] is proved only for primes π, a similar argument =-=[20]-=- shows that the analogous graphs defined for powers of π have the same properties. If π is a prime congruent to 1 modulo 4, p is a quadratic residue modulo π, and l is an integer, denote by Pl the fac... |

4 |
A note on lower bounds
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(Show Context)
Citation Context ...ce δ and of rate R ≥ RGV(δ), where q RGV(δ) ∆ = 1 − Hq(δ) , (1) and Hq(x) ∆ = −x · logq x − (1 − x) · logq(1 − x) + x · logq(q − 1), 0 ≤ x ≤ 1 − 1. Furthermore, q the seminal works of Tsfasman et al. =-=[10, 13, 25]-=- show the existence of good code sequences beyond the Gilbert-Varshamov bound for q ≥ 46. A code sequence S = {Ci} ∞ i=1 over an alphabet Σ is called constructive if there exists an algorithm that com... |

4 |
Justesen’s construction — the low-rate case
- Weldon
- 1973
(Show Context)
Citation Context ...operations, where lim 1 δ→1− c(δ) = ∞. Hence, such a code sequence family is q non-uniformly constructive. The exponential search is avoided in Justesen codes [9] and in constructions derived thereof =-=[22, 23, 24, 27]-=-, where the inner codes exhaust all members of Wozencraft’s ensemble of randomly shifted codes. Justesen’s construction is also “explicit” in the sense that once the rates of the inner and outer codes... |

3 | A new class of asymptotically good codes beyond the Zyablov bound - Sugiyama, Kasahara, et al. - 1978 |

3 |
Sur les courbes algébriques et les variéstés qui sèn déduisent, Actualités Sci
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Citation Context ...is obtained by applying results of Eichler and Igusa concerning the Ramanujan conjecture. Eichler’s proof relies on Weil’s famous theorem known as the Riemann hypothesis for curves over finite fields =-=[26]-=-. Therefore, for 7severy fixed π, the family {G(p, π)}p is an optimal set of pseudo-random graphs as it attains the bound (6). Although the construction given in [14] and [16] is proved only for prime... |

2 |
Brigand, On computational complexity of some algebraic curves over finite fields
- Le
- 1986
(Show Context)
Citation Context ...cific inner codes, yield a uniformly constructive family that lies above the Zyablov bound. However, since the time complexity of finding the generator matrices of these codes is proportional to n 32 =-=[5]-=-, they can hardly be called constructive from any practical perspective. Apart from this construction, (3) yields the best uniformly constructive family for sufficiently low rates (i.e., when δ is clo... |

2 |
A modification of the constructive asymptotically good codes of Justesen for low rates
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(Show Context)
Citation Context ...operations, where lim 1 δ→1− c(δ) = ∞. Hence, such a code sequence family is q non-uniformly constructive. The exponential search is avoided in Justesen codes [9] and in constructions derived thereof =-=[22, 23, 24, 27]-=-, where the inner codes exhaust all members of Wozencraft’s ensemble of randomly shifted codes. Justesen’s construction is also “explicit” in the sense that once the rates of the inner and outer codes... |

2 |
Superimposed concatenated codes
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- 1980
(Show Context)
Citation Context ...operations, where lim 1 δ→1− c(δ) = ∞. Hence, such a code sequence family is q non-uniformly constructive. The exponential search is avoided in Justesen codes [9] and in constructions derived thereof =-=[22, 23, 24, 27]-=-, where the inner codes exhaust all members of Wozencraft’s ensemble of randomly shifted codes. Justesen’s construction is also “explicit” in the sense that once the rates of the inner and outer codes... |

2 |
Some results on the problem of constructing asymptotically good error-correcting codes
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(Show Context)
Citation Context ...o be strictly smaller than 1 − . Therefore, Justesen 2 q codes do not comply with requirement (ii) of uniform constructiveness. The same holds also for some other known improvements on Justesen codes =-=[23, 28]-=-. Uniformly constructive families of codes over GF (q) were obtained by Weldon [27] and Sugiyama et al. [22, 24], where the outer Reed-Solomon codes were replaced by much longer codes over GF (q m ), ... |