## Abstract

People rely on the ability to transmit information over channels of communication that aresubject to noise and interference. This makes the ability to detect and recover from errorsextremely important. Coding theory addresses this need for reliability. A fundamentalquestion of coding theory is whether and how we can correct the errors in a message thathas been subjected to interference. One answer comes from structures known as errorcorrecting codes.A well studied parameter associated with a code is its covering radius. The coveringradius of a code is the smallest radius such that every vector in the Hamming space of thecode is contained in a ball of that radius centered around some codeword. Covering radiusrelates to an important decoding strategy known as nearest neighbor decoding.The multicovering radius is a generalization of the covering radius that was proposed byKlapper [11] in the course of studying stream ciphers. In this work we develop techniques forfinding the multicovering radius of specific codes. In particular, we study the even weightcode, the 2-error correcting BCH code, and linear codes with covering radius one.We also study questions involving the complexity of finding the multicovering radius ofcodes. We show: Lower bounding the m-covering radius of an arbitrary binary code is NPcompletewhen m is polynomial in the length of the code. Lower bounding the m-coveringradius of a linear code is Î£p2-complete when m is polynomial in the length of the code. IfP is not equal to NP, then the m-covering radius of an arbitrary binary code cannot beapproximated within a constant factor or within a factor nÏµ, where n is the length of thecode and Ïµ andlt; 1, in polynomial time. Note that the case when m = 1 was also previouslyunknown. If NP is not equal to Î£p2, then the m-covering radius of a linear code cannot beapproximated within a constant factor or within a factor nÏµ, where n is the length of thecode and Ïµ andlt; 1, in polynomial time.

Original language | American English |
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Journal | Default journal |

State | Published - Jan 1 2005 |

Externally published | Yes |

## Keywords

- Coding Theory|Complexity|Covering Radius|Multicovering Radius|Approximation Complexity