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 D2n, 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 D2nv. 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 D2nv
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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