The Characterization of Vectors in Rⁿ with the Haar Property

Author Information

Terrence Kelleher

Abstract/Description

Introduction:
As modern communication becomes more digital than ever, infrastructure tries to maintain an equal pace. Unfortunately, this is not always possible. Therefore, computer scientists and mathematicians alike have endeavored to invent ways to store, send and retrieve data, even if transmitted signals are severely damaged. One such way is by using what is called an error-correcting code, or an ECC. An ECC is a method of encoding information such that a signal that possesses a message can be significantly altered and still be decoded. There are many different types of ECCs. The type that is of interest in this paper depends upon mathematical structures for resilience to inadvertent alteration of a message.

In this paper we will consider the mathematical structure behind one type of error correcting code. Its purpose is to introduce redundancy in the way information is coded so that losses, noises, and erasures to transmitted signals do not prevent accurate reconstruction of information. We represent this process by first letting a given signal be represented by a vector v in an n-dimensional Euclidean space denoted ℝⁿ. It is worth noting that such vectors will be represented in column-form. Secondly, we consider a representation of the Dihedral group, denoted D_2n, which will act in a linear fashion by matrix-vector multiplication on ℝⁿ. Next, we say for a fixed vector v, the set of all vectors obtained by acting on v is called an orbit of v and is denoted D_2nv. Finally, we say a vector v has the Haar property if, and only if, every subset of the orbit of v of cardinality n is a basis for the vector space ℝⁿ. In this paper, we give a characterization of vectors in ℝ³ that satisfy the Haar property, and work for similar results in ℝ⁵. It has been shown that in the ℝ⁴ case, we cannot find the Haar property when examining n vector subsets. Therefore we modify the Haar property by looking for n − 1 as linearly independent subsets of D_2nv

Note on the Author

Terrence Kelleher is graduating in May 2016 from Bridgewater State University with a Bachelor’s of Science in Mathematics with a concentration in Pure Mathematics and a minor in Statistics. His research project was made possible by the Adrian Tinsley Program summer research grant in 2015. Terrence and his mentor Dr. Shannon Lockard (Mathematics) answered a question provided by Dr. Vignon Oussa, also of the BSU Department of Mathematics. Terrence presented this research at the 2016 Joint Mathematics Meetings in Seattle, Washington, an opportunity afforded to 200 students chosen from a large pool of applicants from across the nation. Terrence has a passion for learning and imparting that passion to others. Therefore he hopes to become a professor someday.

Recommended Citation

Kelleher, Terrence (2016). The Characterization of Vectors in Rⁿ with the Haar Property. Undergraduate Review, 12, 72-79.
Available at: https://vc.bridgew.edu/undergrad_rev/vol12/iss1/14

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Articles published in The Undergraduate Review are the property of the individual contributors and may not be reprinted, reformatted, repurposed or duplicated, without the contributor’s consent.