Topic
Polynomial code
About: Polynomial code is a research topic. Over the lifetime, 2026 publications have been published within this topic receiving 30666 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: A family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality are presented.
Abstract: A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most $r$) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter $r$ is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over $r$ points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data ("hot data").
582 citations
TL;DR: The nearest-neighbor decoding problem for LCD codes is shown to reduce to the problem: given a word in C⊥, find the nearest codeword in C.
523 citations
TL;DR: It is shown that the problem of computing the minimum distance of a binary linear code is NP-hard, and the corresponding decision problem isNP-complete.
Abstract: It is shown that the problem of computing the minimum distance of a binary linear code is NP-hard, and the corresponding decision problem is NP-complete. This result constitutes a proof of the conjecture of Berlekamp, McEliece, and van Tilborg (1978). Extensions and applications of this result to other problems in coding theory are discussed.
512 citations
TL;DR: A class of decoding algorithms using encoding-and-comparison is considered for error-correcting code spaces and it is suggested on operational grounds that it may prove most useful in those cases where m is relatively large with respect to the code length n.
Abstract: A class of decoding algorithms using encoding-and-comparison is considered for error-correcting code spaces. Code words, each of which agrees on some information set for the code with the word r to be decoded, are constructed and compared with r . An operationally simple algorithm of this type is studied for cyclic code spaces A . Let A have length n , dimension k over some finite field, and minimal Hamming distance m . The construction of fewer than n^2/2 code words is required in decoding a word r . The procedure seems to be most efficient for small minimal distance m , but somewhat paradoxically it is suggested on operational grounds that it may prove most useful in those cases where m is relatively large with respect to the code length n .
496 citations
TL;DR: The spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space) and the two sets of integers are related by a system of linear equations.
Abstract: A systematic code of word length n is a subspace of the vector space of all possible rows of n symbols chosen from a finite field. The weight of a vector is the number of its nonzero coordinates; clearly any given code contains a certain finite number of vectors of each weight from zero to n. This set of integers is called the spectrum of the code, and very little is known about it, although it appears to be important both mathematically and as a practical means of evaluating the error-detecting properties of the code. In this paper it is shown that the spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space). In fact the two sets of integers are related by a system of linear equations. Consequently there is a set of conditions which must be satisfied by the weights which actually occur in a systematic code. If there is enough other information about the code, it is possible to use this result to calculate its spectrum.
475 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 1 |
| 2024 | 1 |
| 2023 | 10 |
| 2022 | 15 |
| 2021 | 13 |
| 2020 | 16 |