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### On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$

Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows  S=\left[\begin{array}{ccccccc} 0 & ...
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Consider the vector space $\mathbb{R}^n$, the standard inner product $\langle \cdot,\cdot \rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}$, and some $0<\epsilon\leq ... 0answers 100 views ### Fully Homomorphic Error Correction? Consider a field$F$. Suppose we have two vectors$a,b\in F^n$, and an invertible matrix$G\in F^{n\times n}$. Let$c\in F^n$be the point-wise product of$a$and$b$, that is,$c_i=a_ib_i$. Let ... 1answer 122 views ### Reference for partial Hadamard matrices Definition. An$m\times n$matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from$\lbrace -1, 1 \rbrace$such that the dot product of each pair of row vectors ... 1answer 152 views ### Lower bound on the dimension of a subspace of$\mathbb Z_2^r$? This question may be very trivial, I apologize if it is so. I have subspace$V\subset \mathbb Z_2^r$with the property that for every choice of a subset$I$of$k$elements in$\{1,2,\dots r\}$, the ... 2answers 230 views ### Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ? Consider system of linear equations Ax=0 over$F_2$(field with two elements {0,1}). Where number of variables is bigger than equations - so we have many solutions$x$. Question How to estimate ... 1answer 715 views ### The generator polynomial of cyclic code Let$q$be a power of prime number$p$and let$F_{q^2}$be a finite field of order$q^2$. Suppose that "-" be a conjugation operation that is defined as follow:$-:F_{q^2} ‎\longrightarrow‎ ...
In coding theory, there are parity-check codes whose parity-check matrices $H$ are generated via column permutations. For instance, the binary LDPC codes constructed in Gallager's 1962 IRE Trans paper ...
I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...