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The theory of error-correcting codes stems from Shannon's 1948 _A mathematical theory of communication_, and from Hamming's 1950 "Error detecting and error correcting codes".

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Let $\mathcal{B}(n,w)$ be the set of all binary vectors of length $n$ and constant weight $w$, i.e., $\mathcal{B}(n,w) = \{ x \in \{0,1\}^n : \mathrm{wt}(x) = w \}$. The Hamming ball of radius $r$ (in ...
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Let $n \ge 1$. A set of vectors $v_1, \ldots, v_m \in \{0,1\}^n$ is called admissible if all pairwise sums $v_i + v_j$ (with $1 \le i \le j \le m$) are distinct. We want to find the number $a(n)$, ...
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Let $\alpha$ be an angle contained in $(0,\pi/2]$ -- to fix ideas, let $\alpha = \pi/3$. Then with $d > 1$ a positive integer, what is a minimal bound $M_d(\alpha)$ for the maximum number of ...
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Problem Statement: Consider two parties, a Sender holding a binary vector $s_1 \in \{0,1\}^d$ and a Receiver holding a binary vector $r_1 \in \{0,1\}^d$, where $d$ is the dimension and $\delta \geq 1$ ...
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Consider the following decision problem, which we will call COMPARE. We are given as input a pair $(V_0, V_1)$ of linear codes in $\mathbb{F}_2^n$, and asked to decide whether $V_0, V_1$ have the same ...
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I would like to know the feasibility of the following linear programming problem related to coding theory. Given a natural number $d$, binary entry matrix $X:=[x(i,j)\in B],\ B:=\{0,1\},\ i\in B^d,\ j\...
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A related post was already put here by me. But no comment received. So I decided to try my luck again here. Let $F = \{0,1,2\}$ be the ternary finite field. A vector $v \in F^n$ is balanced if each of ...
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Let $C\subseteq\mathbb{F}_2^n$ be a linear code and let $P$ be the corresponding weight enumerator polynomial. That is, $$P(x)=a_nx^n+\cdots+a_1x+a_0$$ where, for $0\leq j\leq n$, we have $a_j:=\#\{v\...
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Let $A$ be an affine subspace of $\mathbb{F}_2^n$. Let $m\leq n$ and $Q_0, Q_1$ be linear maps $\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$. Consider the following decision problem: Decide whether or not ...
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Here is an interesting inequality that is simulated to be correct but I can not prove it. Can someone helps me? The question is that: For any binary vector $\mathbf{y}\in\{0,1\}^n$, prove that \begin{...
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$\;\;\;\;$ Fix a prime $p$, fix a power $q:=p^\aleph$, and consider the action of the Frobenius ring automorphism $\Psi:\vec{x}\mapsto\vec{x}^p$ on the product ring $\Bbb F_{q^m}^{(n)}:=\Bbb F_{q^m}\...
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Consider a binary sequence $$\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right ), \quad x_{i}\in\left \{ 0, 1 \right \}$$ and suppose that the total number of $1$’s in the sequence is known. ...
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I will first explain this method in detail using a trivial example, then I will give examples of known nonlinear codes. 1. Trivial example. 1.a. Consider a linear code with a generating matrix ...
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guys, I'm reading the book `Channel Codes:Classical and Modern' by W.E.Ryan and Shu Lin, in paper 15, there is an figure which gives out the hard and soft capacities curve for BI-AWGN channel, as ...
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Basically my question is the following. Suppose $\mathcal{H}$ is a collection of finite subsets of the natural numbers (containing at least one non-empty set) closed under symmetric difference and ...
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Radix economy concerns itself with the efficiency of encoding numbers. For positional number systems that use a fixed base, base three is the most efficient choice among the integers, and $e$ is the ...
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Related to this question. I wish to compute $x^n \bmod g(x)$ in $\mathbb{F}_2[x]$ for some fixed and known upfront $g$. This problem pops up in computing the 'pure' CRC function of a bit sequence of ...
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Assume that the natural numbers have been colored with two colors: lavender and periwinkle. You don't know the coloring. You may sample as many (possibly overlapping) sets of size $n$ as you would ...
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Cross posting from MSE. I think it might be a good fit here. Given a linear binary code $C$, the $r$-th generalized Hamming weight $d_{r}(C)$ is the minimal support size of an $r$-dimensional subcode ...
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$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two subshifts of finite type and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any homeomorphism $\...
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Let $\mathbb{F}_p\left[S_n\right]$ be the group algebra of the symmetric group $S_n$ over the finite field $\mathbb{F}_p$. One can define an "inner product" in the usual way: $$\langle x,y \...
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Let $S$ be a subset of vectors in $\mathbb{F}_2^{3n}$ having Hamming weight $n$. Suppose that $S$ contains $m$ pairs of vectors having disjoint supports (that is, they are at Hamming distance $2n$ ...
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Basically, I'm trying to figure out the name of the thing I want to look up. All the terms I've looked up so far have been related, but not close enough to be useful. I'm trying to find bit strings ...
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I am trying to find out the maximum-sized subset $S\subseteq \{0,1\}^n$ of $n$-bit strings that does not span $\mathbb{R}^n$. It is easy to show that $S$ has size at least $2^{n-1}$ when $S$ exactly ...
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Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$....
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In his thesis (1973), P. Delsarte defines a duality construction for association schemes. Nevertheless, this duality construction works only if some special regularity condition is satisfied. I find ...
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The Gilbert-Varshamov bound provides a lower bound for codes of length $n$ with minimum pairwise distance (say $\frac{n}8$). If we wish for the codes to also have pairwise distances bounded above (say ...
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If you have a binary single-error correcting code with n-bit codewords, then it is the case that taking only a fixed n-1 out of the n bits gives an “approximate” code with the property that, for any ...
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In this question about perfect 2 error correcting codes on the Open Problem Garden, it is stated that: Recent research activity has discovered a large number of previously unknown perfect 1-error ...
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Given $k\ge 2$ and an additive set $S$ (understood to live some implicit group $G$), define $$\Delta_k(S) := \left\{ d \in G: \bigcap_{i=1}^k (S+i\cdot d) \neq \emptyset \right\} $$(i.e., this is the ...
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Hamming-distance-4 binary codes have a very direct relationship to sphere packings. That's because we can identify the codewords with the cosets of $\mathbb{Z}^n/(2\mathbb{Z})^n$, and Hamming-distance-...
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I am a little confused with the relationship between various bounds for error correcting codes. Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically? That is, is ...
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This question is motivated by section 15.1 (Codes) of Alon and Spencer's The probabilistic method. Fix $\alpha<\frac{1}{2}$ and for each $n\in\mathbb{N}$ let $\{0,1\}^n$ be the length $n$ binary ...
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Let $n$ be a large positive integer. Sample $N \ge 2$ points $x_1,\ldots,x_N$ iid from the uniform distribution on the $n$-dimensional hypercube $\{0,1\}^n$. Define the gap $\delta_{N,n} := \min_{i \...
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Let $q = p^s$ and $r = q^m$, where $p$ is a prime, $s$ and $m$ are positive integers. Let $N>1$ be an integer dividing $r - 1$, and put $n = (r - 1)/N$. Let $\alpha$ be a primitive element of $\...
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We are considering the following problem: Given an integer $n$ and a sequence of integers $r_i,\ 1\le i\le n$, with $0\le r_i\le n-1$ does there exists a symmetric matrix $A$ such that the diagonal ...
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People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...
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I'm new to coding theory but would like to ask the following question: Let $C$ be a linear $q$-ary code over $\mathbb{F}_q$ of length $n$ and dimension $k$ where $\mathbb{F}_q$ is a finite field with $...
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sorry for informals but is my first post. In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition: $\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
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I am interested in the maximal number of $k$-subsets of an $n$-set such that any two subsets meet in at most $(k-2)$ points. I found that for $k=3$ and $k=4$, we have the sequences http://oeis.org/...
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Let $V$ be an $n$-dimensional vector space over $\mathbb{F} = \mathbb{Z} / (p)$, the field of $p$ elements, $p$ a prime, with $\{v_1, \dotsc, v_n \}$ a basis for $V$. An element $x \in V$ is called &...
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Let $\chi_X:\{-1,1\}^n\to \{0,1\}$ be the characteristic function of a subset $X\subseteq \{-1,1\}^n$, which is randomly drawn from all subsets with exactly $k$ elements. Is the support of the Fourier ...
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Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property $$AA^T=(n-1)I+J$$ where $I$ is the $n \times n$ identity matrix and $J$ ...
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Consider a $d$-dimensional linear subspace $V\subseteq\mathbb{F}_p^n$, and its intersection with subcubes of form $S_1\times\cdots\times S_n$, where $S_1,\ldots,S_n$ are arbitrary size-$s$ subsets of $...
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We have a long message $m$ to encode. The message is composed of a set of symbols $\{s_i\}$. Let $p_i$ denote the number of appearance of $s_i$ in $m$. We seek to find a prefix-free code for each $s_i$...
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Consider a list of all the subsets of $\{1,\ldots,n\}$, in any order. Using binary notation, one such list for $n=3$ is $$ 010, 100, 110, 011, 000, 111, 001, 101. $$ Now consider the Hamming distance ...
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In a paper on a proximity test for Reed-Solomon codes the authors state an "extremely optimistical" conjecture on the list decodability of Reed-Solomon codes (over prime fields $\mathbb F_q$)...
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A parity check matrix for a binary linear code is a matrix $H$ (with linearly independent rows) such that the (right) kernel of $H$ is the codespace. Clearly we can replace any row $r_i$ by $r_i + \...
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Given a set $\cal N$ of $N$ objects, we seek to attribute a code, i.e., a binary sequence, to each of them to achieve the following objective of being capable to select any subset ${\cal S}\subseteq {\...
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Context: roughly speaking the construction of a error correcting code is a problem to choose some subset in a metric space, such that the points of the subset pairwise as far-distant as possible. ...
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