## Algorithmic Issues in Coding Theory (1997)

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Citations: | 6 - 0 self |

### BibTeX

@MISC{Sudan97algorithmicissues,

author = {Madhu Sudan},

title = {Algorithmic Issues in Coding Theory},

year = {1997}

}

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### Abstract

The goal of this article is to provide a gentle introduction to the basic definitions, goals and constructions in coding theory. In particular we focus on the algorithmic tasks tackled by the theory. We describe some of the classical algebraic constructions of error-correcting codes including the Hamming code, the Hadamard code and the Reed Solomon code. We describe simple proofs of their error-correction properties. We also describe simple and efficient algorithms for decoding these codes. It is our aim that a computer scientist with just a basic knowledge of linear algebra and modern algebra should be able to understand every proof given here. We also describe some recent developments and some salient open problems.

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Citation Context ...hat a theoretical computer scientist be comfortable with the methods of this eld | and this is the goal of this article. A reader interested in further details may try one of the more classical texts =-=[2, 11, 17]-=-. Also, the article of Vardy [18] is highly recommended for a more detailed account of progress in coding theory. The article is also rich with pointers to topics of current interest. 2 Linear Codes W... |

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Citation Context ...hat a theoretical computer scientist be comfortable with the methods of this eld | and this is the goal of this article. A reader interested in further details may try one of the more classical texts =-=[2, 11, 17]-=-. Also, the article of Vardy [18] is highly recommended for a more detailed account of progress in coding theory. The article is also rich with pointers to topics of current interest. 2 Linear Codes W... |

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Citation Context ...lso describe the algorithmic issues motivated by these combinatorial objects and try to provide some solutions (and summarize the open problems). (We assume some familiarity with algebra of nite elds =-=[10, 19]-=-.) Before going on to these issues, we once again stress the importance of the theory of errorcorrecting codes and its relevance to computer science. The obvious applications of error-correcting codes... |

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Citation Context ...ther class of codes, constructed by combinatorial means, for which bounded distance decoding for some t d can be performed in polynomial time. These are the expander codes, due to Sipser and Spielman =-=[14]-=- and Spielman [15]. The results culminate in a code with very strong | linear time (!!!) | encoding and bounded distance decoding algorithms. In addition to being provably fast, the algorithms for the... |

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Citation Context ...hat a theoretical computer scientist be comfortable with the methods of this eld | and this is the goal of this article. A reader interested in further details may try one of the more classical texts =-=[2, 11, 17]-=-. Also, the article of Vardy [18] is highly recommended for a more detailed account of progress in coding theory. The article is also rich with pointers to topics of current interest. 2 Linear Codes W... |

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Citation Context ...des (again using the rst de nition given here of Reed Solomon codes). The distance property of the multivariate polynomial codes follow also from the distance property ofmultivariate polynomials (cf. =-=[5, 13, 21]-=-). Lemma 10. For integers m; l and q with l<q, the code Cpoly;m;l;q is an [n; k; d]q code with n = q m , k = , m+l m and d =(q,l)qm,1 . 3 This identity is obtained as follows: Recall that Fermat's lit... |

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Citation Context ...in exponential time. One may wonder if this exponential time behavior is inherent to the decoding problem. In perhaps the rst \complexity" result in coding theory, Berlekamp, McEliece and van Til=-=borg [4]-=- present the answer to this question. Theorem 16 [4]. The Maximum likelihood decoding problem for general linear codes is NP-hard. There are two potential ways to attempt to circumvent this result. On... |

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Citation Context ...lynomial-time decoding algorithm that corrects more errors - say uptot< (1, )d where is some xed constant. However no such algorithm is known for all possible values of (n; k; d = n , k). Recently, in=-=[16]-=-, we presented an algorithm which does correct up to (1 , )d errors, provided k=n ! 0. This algorithm was inspired by an algorithm of Welch and Berlekamp [20, 3] for decoding Reed Solomon codes. This ... |

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Citation Context ...le to encode the codes faster, in time O(n log c n) for some constant c. However till recently no asymptotically good code was known to be encodable in linear time. In a recent breakthrough. Spielman =-=[15]-=- presented the rst known code that is encodable in linear time. We will discuss this more in a little bit. The next obvious candidate problem is the decoding problem. Once again it is clear that if th... |

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Citation Context ... values of (n; k; d = n , k). Recently, in[16], we presented an algorithm which does correct up to (1 , )d errors, provided k=n ! 0. This algorithm was inspired by an algorithm of Welch and Berlekamp =-=[20, 3]-=- for decoding Reed Solomon codes. This algorithm is especially clean and elegant. Our solution uses similar ideas to correct even more errors and we present this next. Notice rst that the decoding pro... |

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Citation Context ...ce decoding problem, one needs to ensure that the number of outputs (i.e., the number codewords within the given bound t) is polynomial in n. Such a bound does exist for the value of t as given above =-=[6,12]-=-, thus raising the hope that this problem may be solvable in polynomial time also. Similar questions may also be raised about decoding multivariate polynomials. In particular, we don't have polynomial... |

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(Show Context)
Citation Context ...des (again using the rst de nition given here of Reed Solomon codes). The distance property of the multivariate polynomial codes follow also from the distance property ofmultivariate polynomials (cf. =-=[5, 13, 21]-=-). Lemma 10. For integers m; l and q with l<q, the code Cpoly;m;l;q is an [n; k; d]q code with n = q m , k = , m+l m and d =(q,l)qm,1 . 3 This identity is obtained as follows: Recall that Fermat's lit... |

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Citation Context ...ents. The existence of such coe cients will determine our choice of m; l. Having determined such a polynomial we will apply the following useful lemma to show that p can be extracted from Q. Lemma 20 =-=[1]-=-. Let Q(x; y) = P i;j qijx i y j be such that qij =0for every i; j with i +(k,1)j >l. Then if p(x) is polynomial of degree k , 1 such that for strictly more than l values of i, yi = p(xi) and Q(xi;yi)... |

45 |
Polynomial factorization 1987–1991
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(Show Context)
Citation Context .... Thus in this case also we have a polynomial time algorithm provided Q can be factored in polynomial time. Fortunately, such algorithms are known, due to Kaltofen [8] and Grigoriev [7] (see Kaltofen =-=[9]-=- for a survey of polynomial factorization algorithms). For k=n ! 0, the number of errors corrected by this algorithm approaches (1 , o(1))n. A more detailed analysis of this algorithm and the number o... |

36 | Algorithmic complexity in coding theory and the minimum distance problem
- Vardy
- 1997
(Show Context)
Citation Context ...comfortable with the methods of this eld | and this is the goal of this article. A reader interested in further details may try one of the more classical texts [2, 11, 17]. Also, the article of Vardy =-=[18]-=- is highly recommended for a more detailed account of progress in coding theory. The article is also rich with pointers to topics of current interest. 2 Linear Codes While all questions relating to co... |

29 |
Bounded distance + 1 soft decision Reed-Solomon coding
- Berlekamp
- 1996
(Show Context)
Citation Context ... values of (n; k; d = n , k). Recently, in[16], we presented an algorithm which does correct up to (1 , )d errors, provided k=n ! 0. This algorithm was inspired by an algorithm of Welch and Berlekamp =-=[20, 3]-=- for decoding Reed Solomon codes. This algorithm is especially clean and elegant. Our solution uses similar ideas to correct even more errors and we present this next. Notice rst that the decoding pro... |

29 |
Fast probabilistic algorithms for veri of polynomial identities
- Schwartz
- 1980
(Show Context)
Citation Context ...es (again using thesrst denition given here of Reed Solomon codes). The distance property of the multivariate polynomial codes follow also from the distance property of multivariate polynomials (cf. [=-=5, 13, 21-=-]). Lemma 6. For integers m; l and q with lsq, the code C poly;m;l;q is an [n; k; d] q code with n = q m , k = m+l m and d = (q l)q m 1 . Proof. The bound on n is immediate. The fact that the number ... |

17 |
Factorization of polynomials over a finite field and the solution of systems of algebraic equations
- Grigoriev
- 1986
(Show Context)
Citation Context ...ost t values of xi. Thus in this case also we have a polynomial time algorithm provided Q can be factored in polynomial time. Fortunately, such algorithms are known, due to Kaltofen [8] and Grigoriev =-=[7]-=- (see Kaltofen [9] for a survey of polynomial factorization algorithms). For k=n ! 0, the number of errors corrected by this algorithm approaches (1 , o(1))n. A more detailed analysis of this algorith... |

14 |
Fast probabilistic algorithms for veri cation of polynomial identities
- Schwartz
- 1980
(Show Context)
Citation Context ...des (again using the rst de nition given here of Reed Solomon codes). The distance property of the multivariate polynomial codes follow also from the distance property ofmultivariate polynomials (cf. =-=[5, 13, 21]-=-). Lemma 10. For integers m; l and q with l<q, the code Cpoly;m;l;q is an [n; k; d]q code with n = q m , k = , m+l m and d =(q,l)qm,1 . 3 This identity is obtained as follows: Recall that Fermat's lit... |

13 | A polynomial-time reduction from bivariate to univariate integral polynomial factorization
- Kaltofen
- 1982
(Show Context)
Citation Context ...xi) 6= yi for at most t values of xi. Thus in this case also we have a polynomial time algorithm provided Q can be factored in polynomial time. Fortunately, such algorithms are known, due to Kaltofen =-=[8]-=- and Grigoriev [7] (see Kaltofen [9] for a survey of polynomial factorization algorithms). For k=n ! 0, the number of errors corrected by this algorithm approaches (1 , o(1))n. A more detailed analysi... |