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2008, Doctor of Philosophy, Ohio State University, Mathematics.

We consider the random system of 2-linear equations over the finite field *GF*(2) whose left hand side corresponds to the random graph *G*(*n*,*p*) and whose right hand side consists of independent Bernoulli random variables with success probability 1/2, assuming that the right hand side is independent of the left hand side.

*G*(*n*,*p*) is the random graph with *n* labeled vertices such that each of all the n(n-1)/2 possible edges is present in the graph independently of all others, with probability p.

We prove that when *G*(*n*,*p*) is at the subcritical phase and |λ| >> n^{1/39}, |λ| = O(n^{1/12-ε}) with a fixed 0<ε<1/12-1/39, the probability of solvability of the random system corresponding to *G*(*n*,*p*) is asymptotic to e^{3/8}

|λ| ^{1/4} n^{-1/12} as *n* goes to infinity. Also, we prove that when *G*(*n*,*p*) is at the critical phase, the probability of solvability of the random system corresponding to *G*(*n*,*p*) is asymptotic to c_{λ} n^{-1/12} as n goes to infinity, where the constant c_{λ} is expressed as a convergent double series depending on λ.

Boris Pittel (Advisor)

Akos Seress (Committee Member)

Saleh Tanveer (Committee Member)

Nicholas Hall (Committee Member)

Akos Seress (Committee Member)

Saleh Tanveer (Committee Member)

Nicholas Hall (Committee Member)

Yeum, J. (2008). Probability of Solvability of Random Systems of 2-Linear Equations over *GF*(2). (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

Yeum, Ji-A. "Probability of Solvability of Random Systems of 2-Linear Equations over *GF*(2)." Electronic Thesis or Dissertation. Ohio State University, 2008. OhioLINK Electronic Theses and Dissertations Center. 25 Feb 2018.

Yeum, Ji-A "Probability of Solvability of Random Systems of 2-Linear Equations over *GF*(2)." Electronic Thesis or Dissertation. Ohio State University, 2008. https://etd.ohiolink.edu/