**20**

votes

**2**answers

1k views

### What is the complexity of this problem?

Recently on Dick Lipton and Ken Regan's blog there was a post about problems of intermediate complexity, that is, NP problems that are harder than P but easier than NP-complete. The main message of ...

**15**

votes

**1**answer

677 views

### Codes, lattices, vertex operator algebras

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention ...

**14**

votes

**1**answer

863 views

### How to keep subsets disjoint?

Given positive integers $n$ and $k\le 2^n$, how to choose a subset $C\subset\{0,1\}^n$ of size $|C|=k$ to maximize the number of pairs $(c_1,c_2)\in C\times C$ with the supports of $c_1$ and $c_2$ ...

**13**

votes

**3**answers

715 views

### Can you cover the Boolean cube {0,1}^n with O(1) Hamming-balls each of radius n/2-c*sqrt(n)?

(where c>0 and the balls need not be disjoint?)
This is an embarrassingly simple question, yet somehow I couldn't find an answer (not even, "this is a well-known open problem") after spending some ...

**10**

votes

**3**answers

575 views

### MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction
Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the ...

**9**

votes

**1**answer

274 views

### An isoperimetric problem on the hypercube

Fix integer $n\ge 1$, and let $E=\{e_1,...,e_n\}$ denote the standard basis of the vector space ${\mathbb F}_2^n$. Thus, for a set $A\subset{\mathbb F}_2^n$, the sumset $A+E:=\{a+e\colon a\in A,\ e\in ...

**7**

votes

**3**answers

806 views

### Hot-topics in error correcting coding related to interesting math. ?

What are topics in error-correcting coding which are related to interesting math. ?
I am primarely interested in nowdays hot topics, but old days topics are also welcome.
Let me try to mention what ...

**7**

votes

**4**answers

1k views

### “sparse graphs are locally tree-like”

I would like to be able to state with confidence that sparse graphs (graphs with small numbers of edges) are locally tree-like (they have few short cycles). Apparently "Sparse graphs are locally tree ...

**7**

votes

**3**answers

595 views

### Error correcting codes - basic question

Hi,
I have a basic question regarding error correcting codes. Suppose I want to encode a finite information $F$ (say a finite string) into a string $x$ of $n$ bits ($n$ can be as large as you want), ...

**7**

votes

**1**answer

2k views

### 12 balls weighing puzzle

In an article describing the twelve balls weighing problem, the author mentions a solution that involves the finite projective plane of order 3, discovered by Rick Wilson. Does anyone know what this ...

**7**

votes

**2**answers

549 views

### Can you hide a letter without losing information?

Consider the following game between Alice and Bob.
$\Sigma$ is a finite nonempty alphabet, $\Delta \notin \Sigma$ denotes
a special symbol, and $k > 0$ is a positive integer constant representing
...

**7**

votes

**2**answers

524 views

### Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$.
There's a ...

**6**

votes

**5**answers

570 views

### covering by spherical caps

Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...

**6**

votes

**3**answers

322 views

### 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 & ...

**6**

votes

**2**answers

378 views

### request sources about self-dual cyclic codes

X. Kai and S. Zhu in the paper "On cyclic self-dual codes", AAECC, vol. 19, pp. 509-525, 2008, at page 510 in line 6 said that, It is well Known that there are no cyclic self-dual codes over $F_q$ ...

**5**

votes

**1**answer

959 views

### binary code with constant hamming distance

I want as many 80-bits words as possible with the constraint that the hamming distance between any couple of words is exactly 40. How many can I generate? Is there a generic formula telling me how ...

**5**

votes

**2**answers

534 views

### Binary codes with large distance

Suppose $C \subset \lbrace 0,1\rbrace^{n}$ is a binary code with distance $\delta * n$, for $1/2 < \delta < 1$. Can $|C|$ be arbitrarily large (if I allow n to be arbitrarily large)? Note that ...

**5**

votes

**2**answers

799 views

### Are algebraic geometry error correcting codes (Goppa codes) “good” ?

Question (informal version): Are algebraic geometry error correcting codes (V.D. Goppa codes) "good" ?
Some details. There is certain construction of error-correcting codes by means of algebraic ...

**5**

votes

**0**answers

159 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 ...

**5**

votes

**0**answers

255 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 ...

**4**

votes

**2**answers

868 views

### Reed-Muller-Codes

Let $F$ be the field with two elements, $V_m=F^{2^m}$.Let $R(r, m)\subset V_m$ be the
binary Reed-Muller Code. Define $R_m:=R(1, m)$. Then the dimension of $R_m$ is
$1+m$ and its minimal distance is ...

**4**

votes

**3**answers

283 views

### Bounded Hamming distance

Definition 1. For each $n\in\mathbb{Z}^+$, the $n$-dimensional Hamming cube is the set of ordered $n$-tuples of $\lbrace 0,1\rbrace$, denoted by $\lbrace 0,1\rbrace ^n$.
Definition 2. The binary ...

**4**

votes

**2**answers

218 views

### Is there a code which corrects corruption of any two bits in a block?

Background
I've just learned a bit about linear codes. Hamming codes have the property that up to one bit in a block can be corrupted, and we still communicate the message correctly. This is done by ...

**4**

votes

**1**answer

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 ...

**4**

votes

**1**answer

194 views

### Best upper bound on rate for q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch which states that the rate $R(\delta)$ corresponding to ...

**4**

votes

**4**answers

618 views

### how to find vector $(\pm 1, \pm 1, \pm 1, … \pm 1)$ which is most close to given vector (r_1,…r_l) ? Is it NP-problem ? What algorithms are available ?

Given:
some vector $R=(r_1...r_l)$ - real numbers,
and a set of distinct vectors with $+1$ or $-1$ coordinates
$$\begin{array}{c} V_1=(c_{1,1} ... c_{1,l}),\\\
V_2=(c_{2,1} ... c_{2,l}),\\\
...

**4**

votes

**2**answers

127 views

### Another formulation of error-correcting coding problem

Consider classic error-correcting problem:
there is finite set $A$ and string $a_1...a_n$, $a_i \in A$ in the begin.
in the end we have $b_1...b_n$. Set places of errors $E = \{i| a_i\not= b_i \}$, ...

**4**

votes

**2**answers

355 views

### A Boolean function that is not constant on affine subspaces of large enough dimension

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 ...

**4**

votes

**1**answer

253 views

### Probability of false decoding with LDPC codes

It is well known that given a binary linear code $C$ under maximum likelihood decoding the probability of false decoding $P_C$ in a Binary Symmetric Channel with cross over probability $p$, in other ...

**4**

votes

**1**answer

217 views

### Most orthogonal lattice basis

Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...

**4**

votes

**1**answer

184 views

### Partial backups

Suppose you have some storage medium of a given size M, and can make some kind of backup on another medium of size B with M > B. You can choose the scheme to determine the contents of the backup.
...

**4**

votes

**1**answer

405 views

### Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?

Since my original posting some ten days ago, I discovered an amazing
example which changed significantly my perception of the problem.
Accordingly, the whole post got re-written now.
The most ...

**4**

votes

**0**answers

104 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

**3**answers

490 views

### Structure of F_p[G], for finite group G ?

Consider group algebra k[G] of finite group G.
If k is alg.closed then every irrep lives there with multiplicity equal to dimension.
(More conceptually as bimodule over GxG it is multiplicity free and ...

**3**

votes

**3**answers

411 views

### Good codes in practice for correcting combination of errors and erasures

In practice, both errors and erasures might be introduced in the channel. Could you point me to some good codes for correcting such combinations. Also what are their correction capabilities?

**3**

votes

**3**answers

1k views

### Computing channel capacities for non-symmetric channels

I'm studying information theory right now and I'm reading about channel capacities.
I know that there are known expressions for computing the capacities for some well known simple channels such as ...

**3**

votes

**1**answer

436 views

### “Codes” in which a group of words are pairwise different at a certain position

I read the following problem, claimed to be in the IMO shortlist in 1988:
A test consists of four multiple choice problems, each with three options, and the students should give an unique answer ...

**3**

votes

**1**answer

198 views

### Generator Matrices of Best Known Linear Codes

Is there a location where one can access generator matrices (not just bounds) of best known linear codes?

**3**

votes

**2**answers

265 views

### Cancellation theorem for lattices

By a lattice, we mean a finitely generated, free $\mathbb{Z}$-module together with a symmetric bilinear form. Typical examples are the hyperbolic lattices $U$ and the root lattices $A_{n}, D_{n}, ...

**3**

votes

**1**answer

286 views

### How many vectors of Hamming weight L in “random” K dimensional subspace of F_2^N ? Or how good/bad is random linear block code ?

Consider linear $N$-dimensional space $F_2^N$.
Consider its $K$ dimensional subspace $V \subset F_2^N$.
Let us calculate $w(k,V,N)$ number of vectors in $V$ of Hamming weight $k$ in $V$.
Since there ...

**3**

votes

**1**answer

237 views

### Special polynomials over finite fields

My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any ...

**3**

votes

**2**answers

2k views

### If graph is tree what can be said about its adjacency matrix ?

Question If graph is tree what can be said about its adjacency matrix ? And vice versa ?
Especially I am interested in case when graph is bipartite graph.
Such graphs are related to ...

**3**

votes

**2**answers

169 views

### Binary codes with upper bound on pairwise distance

A fundamental problem in coding theory is:
Given positive integers $k\le n$, determine or bound the maximal cardinality of a set $S\subset \mathbb{F}_{2}^{n}$ such that $\forall x,y\in S: x\ne ...

**3**

votes

**2**answers

128 views

### Techniques for showing optimality of given packing

There are some natural packing problems that have been asked in mathematics. Some of them are:
1)How many balls can be placed with in a cube?
2)How many equidistant points can be place on the ...

**3**

votes

**1**answer

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 ...

**3**

votes

**1**answer

240 views

### If “force” is periodic does it imply “velocity” is periodic ? (or decoding tail-bited conv. codes)

I'll try to translate certain problem about convolutional codes to more common language of ODE, hope my translation is correct, but welcome to criticize.
Consider two given functions periodic ...

**3**

votes

**1**answer

256 views

### The chromatic number of a Hamming-related graph

For integer $1\le k\le n$, let ${\overline H}_n^k$ denote the complement of
the $k$-th power of the Hamming graph on the vertex set ${\mathbb
F}_2^n$; that is, two vectors from ${\mathbb F}_2^n$ are ...

**3**

votes

**2**answers

174 views

### Are there subsets L in R^n such that it is “easy to find” closest point in L to a given P in R^n ? Vague question motivated by error-correcting codes

Math Motivation: consider LINEAR subspace $L$ in $R^n$ and given vector $E$ in $R^n$, then it is easy to find a closest vector $S \in L$ to $E$ - just ortogonal projection.
Question Are they some ...

**3**

votes

**0**answers

80 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

166 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 $m$.
$n(m)$ ...