Approximating the Maximum Weight of Linear Codes is <I>APX</I>-Complete

Toshiya ITOH

Approximating the Maximum Weight of Linear Codes is APX-Complete

Toshiya ITOH

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The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E83-A No.4 pp.606-613

Publication Date: 2000/04/25

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Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

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Toshiya ITOH, "Approximating the Maximum Weight of Linear Codes is APX-Complete" in IEICE TRANSACTIONS on Fundamentals, vol. E83-A, no. 4, pp. 606-613, April 2000, doi: .
Abstract: The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e83-a_4_606/_p

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@ARTICLE{e83-a_4_606,
author={Toshiya ITOH, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Approximating the Maximum Weight of Linear Codes is APX-Complete},
year={2000},
volume={E83-A},
number={4},
pages={606-613},
abstract={The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.},
keywords={},
doi={},
ISSN={},
month={April},}

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TY - JOUR
TI - Approximating the Maximum Weight of Linear Codes is APX-Complete
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 606
EP - 613
AU - Toshiya ITOH
PY - 2000
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E83-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 2000
AB - The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.
ER -