31 lines
1.6 KiB
TeX
31 lines
1.6 KiB
TeX
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% -*- root: main.tex -*-
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\section{Exact Results}
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\label{sec:exact}
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We turn to computing the exact values of $\sum\limits_{\wVec, \wVecPrime \in \pw} \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVecPrime}.$ Starting with the term $\gIJ = \sum\limits_{\wVecPrime \in \pw}\sketchPolarParam{\wVecPrime}$, by the definition of $\sketchPolar$ and the property of associativity in addition, we can break the sum into
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\begin{equation*}
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\gIJ = \sum_{\substack{\wVecPrime \in \pw \st\\
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\sketchPolarParam{\wVecPrime} = 0}} 1 + \sum_{\substack{\wVecPrime \in \pw \st\\
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\sketchPolarParam{\wVecPrime} = 1}} -1.
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\end{equation*}
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Setting the terms to $T_1 = \sum_{\substack{\wVecPrime \in \pw \st\\
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\sketchPolarParam{\wVecPrime} = 0}} 1$ and $T_2 = \sum_{\substack{\wVecPrime \in \pw \st\\
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\sketchPolarParam{\wVecPrime} = 1}} -1$ and fixing $\buck$ to a specific value, gives a system of linear equations for each term. It is a known result for a consistent multiplication that the number of solutions are $| \kDom |^{\numTup - rank(\matrixH')}$. This gives us an exact calculation for both terms,
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\begin{align*}
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T_1 \in \{0, 2^{\numTup - rank(\matrixH')}\},\\
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T_1 \in \{0, 2^{\numTup - rank(\matrixH')}\}.
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\end{align*}
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\subsection{Algorithm for $\gIJ$}
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\begin{algorithmic}
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\If {$\matrixH' \cdot \wVec = j^{(0)}$ is consistent}
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\If {$\matrixH' \cdot \wVec = j^{(1)}$ is consistent}
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\State $\gIJ = 0$
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\Else
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\State $\gIJ = 2^{computeRank(\matrixH')}$
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\EndIf
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\ElsIf{$\matrixH' \cdot \wVec = \buck^{(1)}$ is consistent}
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\State $\gIJ = 2^{computeRank(\matrixH')}$
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\Else
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$\gIJ = 0$
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\EndIf.
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\end{algorithmic}
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