#### 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 ℝ^{n}. 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 ℝ^{n}. 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*_{2n}**v. **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 ℝ^{n}. In this paper, we give a characterization of vectors in ℝ^{3} that satisfy the Haar property, and work for similar results in ℝ^{5}. It has been shown that in the ℝ^{4} 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*_{2n}**v **

#### Recommended Citation

Kelleher, Terrence
(2016).
The Characterization of Vectors in R^{n} 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.