**1**

vote

**1**answer

45 views

### Parity-check matrix for code with variable block size and minimum distance

Consider a linear error-correcting code with symbols in $GF(q)$, with codewords of length $k$ generated from messages of length $n$ and minimum distance $d+1 = k-n+1$. In the cases of interest, $q = ...

**1**

vote

**1**answer

82 views

### Coding for channels with concentrated error

Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$?
We know that if a channel with error flips (in the ...

**9**

votes

**0**answers

177 views

### Maximum number of vectors in a hypercube satisfying given conditions

$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...

**9**

votes

**0**answers

313 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

**5**

votes

**0**answers

196 views

### When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by
$$
L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}.
$$
A ...

**3**

votes

**0**answers

140 views

### On a problem of sphere-packing for Reed-Solomon codes

Suppose we have an $[n, k+1, n-k]$ Reed Solomon code $\mathcal C$ over $\mathbb F_q$, where $n-k=d$ is the minimum distance, and suppose that $d=2t+1$. We know that for every $r \in \mathbb F_q^n$ the ...

**3**

votes

**0**answers

102 views

### Applications of list decoding

This is citation from http://en.wikipedia.org/wiki/List_decoding:
Algorithms developed for list decoding of several interesting code families have found interesting applications in computational ...

**3**

votes

**0**answers

98 views

### Certain subgroup of automorphism groups of binary codes

Suppose that $C$ is an binary linear code of length $n$ and dimension $k$ (i.e. it's a $k$-dimensional linear subspace of $\mathbb{F}_2^n$). As usual, the automorphism group of $C$ is the subgroup of ...

**3**

votes

**0**answers

127 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 ...

**3**

votes

**0**answers

178 views

### Matrix where every subset of rows has maximal rank

I am looking for a class of matrices $M(n(m), m, k(m), \phi)$ with the following properties:
M is $n \times m$ where $n(m) > m$.
Every subset of rows of size $k$ has (maximal) rank $m$.
$n(m)$ ...

**2**

votes

**0**answers

160 views

### PRNG and coding theory

Let $k, n \in \mathbb{N}$, $k = (1 - \epsilon)n$ where $1 >\epsilon > 0$.
I want to find $f: \{0,1\}^k \to \{0, 1\}^n$
such that:
1) $f(a) \not= f(b)$ if $a \not=b $
2) for any $x \in ...

**2**

votes

**0**answers

76 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

**2**

votes

**0**answers

109 views

### Looking for Camion - Abelian codes

I am looking for a copy of the old report "Paul Camion - Abelian codes", Technical Report 1059, University of Wisconsin 1971. I have asked Paul himself, but he could not help me. Anyone out there has ...

**2**

votes

**0**answers

127 views

### Find polynoms f,g such that for any polynom p(x): |fp|+|gp|>= |f|+|g| ? Where |*| is number of non-zero monoms.

How to construct examples (describe all) polynoms f,g such that for any polynom p(x) not equal to zero: |fp|+|gp|>= |f|+|g| ? Where |*| is number of non-zero monoms (=Hamming weight, by the way it ...

**2**

votes

**0**answers

190 views

### What odd-length binary codes have Hamming weights restricted to be multiples of eight?

Let $G$ be a $k$ by $n$ binary matrix with row vectors $\lbrace \vec{x}_j {\rbrace} _{j=1}^k$. We can interpret $G$ as a generator matrix of a linear $[n,k]$ code $\cal{C}$ whose codewords consist of ...

**1**

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**0**answers

173 views

### Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote ...

**1**

vote

**0**answers

212 views

### Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric

Slight extension of cross posting from
http://cstheory.stackexchange.com/questions/7408/lee-metric-gilbert-varshamov-and-hamming-bounds-for-larger-relative-distance-rang (closed there)
The following ...

**0**

votes

**0**answers

101 views

### Minimum-size spanning group using hamming distance

For given natural numbers $N$ and $D$, we will define $F$ as all the non-negative integers less than $2^N$. A group $A \subseteq F$ will be "spanning", if and only if every number $x$ in $F$ satisfies ...

**0**

votes

**0**answers

179 views

### Asymtotic Complexity Analysis using logarithms and binomial coefficients

On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} ...

**0**

votes

**0**answers

317 views

### Adjacency matrices of graphs as parity check matrices of error correcting codes

Consider bipartite graph.
Consider its adjacency matrix.
It will have a form
0 A^t
A 0
Take matrix $A$.
Consider the null-space $L$ of $A$ over $F_2^N$.
Question Can we say something about the ...

**0**

votes

**0**answers

105 views

### A search for optimal order ideals

At the behest of Gerhard Paseman I'll describe the problem that I alluded to in
name for a partial order.
Let $M = M(\infty)$ denote the set of all finite subsets of the positive integers ...