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Browsing College of Science by Author "Agnarsson, Geir"
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Item Classes of High-Performance Quantum LDPC Codes From Finte Projective Geometries(2012-10-05) Farinholt, Jacob M.; Farinholt, Jacob M.; Agnarsson, GeirDue to their fast decoding algorithms, quantum generalizations of low-density parity check, or LDPC, codes have been invesitgated as a solution to the problem of decoherence in fragile quantum states [1, 2]. However, the additional twisted inner product requirements of quantum stabilizer codes force four-cycles and eliminate the possibility of randomly generated quantum LDPC codes. Moreover, the classes of quantum LDPC codes discovered thus far generally have unknown or small minimum distance, or a fixed rate (see [3, 4] and references therin). This paper presents several new classes of quantum LDPC codes constructed from finite projective planes. These codes have rates that increase with the block length n and minimum weights proportional to n1=2. For the sake of completeness, we include an introduction to classical error correction and LDPC codes, and provide a review of quantum communication, quantum stabilizer codes, and finite projective geometry.Item Matrix Algebras: Equivalent Ring Relations and Special Presentations(2017) Mendelson, Samuel Stephen; Mendelson, Samuel Stephen; Agnarsson, GeirRecognizing when a ring is a matrix ring is of significant importance in the study of algebra. A well-known result in noncommutative ring theory states that a ring $R$ is a matrix ring if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$; in which case $R\cong M_2(S)$ for some $S$ that can be completely described in terms of these matrix units. However, finding and verifying a set of matrix units can be difficult. A more recent result states that a ring $R$ is an $(m+n)\times(m+n)$ matrix ring if, and only if, it contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$, in which case $R\cong M_{m+n}(S)$ for some $S$. Under these relations very little is known about the structure of $S$. In this dissertation we investigate algebras over a commutative ring $A$ (or a field $k$) with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We develop results about the structure of these algebras and their underlying rings when $\gcd(i,j)=1$ and then generalize these results for all $i$ and $j$. We then present some interesting examples demonstrating the more subtle characteristics of these algebras. Finally, we develop techniques to see when these algebras can be mapped to $2\times 2$ matrix rings.Item Zero Sum Properties in Groups(2014-03-09) Wu, Angelina A.; Wu, Angelina A.; Agnarsson, GeirA conjecture by Erdős and Lemke in elementary number theory goes as follows: If d is a divisor of n and we have d divisors of n, say a1,...,ad, not necessarily distinct, can we always find a subsequence among them such that their sum is (i) divisible by d, and (ii) at most n? -This was proved by Lemke and Kleitman to be indeed the case. They also noted that an equivalent version of their theorem, stated in terms of the additive cyclic group G = Zn is as follows: Every sequence of n elements of G, not necessarily distinct, contains a subsequence g1,...,gk such that g1+...+gk = 0 and Σki=1/|gi|<=1. This has been shown to be correct for every nite abelian group G. Hence a natural question is therefore if this holds true for any finite group G. By the aid of a computer this has been verified for all solvable groups of order 21 or less, but it is still not known whether it holds for all finite groups. - This paper proves that some well-known non-abelian groups have this property, for example the alternating groups An and symmetric groups Sn for n = 3,4,5,6, the dihedral group Dn for every n and the dicyclic group of every order. Some speculations on possible plan of attack for Sn for larger n are finally discussed.