**1**

vote

**1**answer

1k views

### What is “automorphism group of an error-correcting code” ?

Here in Wikipedia is written: "The automorphism group of the binary Golay code is the Mathieu group M23."
What is "automorphism group of code" ?
PS
Are there other nice examples of relation ...

**2**

votes

**2**answers

304 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 ...

**5**

votes

**2**answers

3k 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 ...

**0**

votes

**0**answers

365 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 ...

**3**

votes

**1**answer

359 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 ...

**7**

votes

**3**answers

650 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), ...

**1**

vote

**1**answer

171 views

### What are the key applications of the MacWilliams identities in coding theory?

The MacWilliams identities relate the weight enumerators of a code and its dual code. I treated the version for linear codes in my combinatorics class, but felt unsatisfied because I didn't have a use ...

**0**

votes

**2**answers

475 views

### Hamming graphs and power series

Let $i$ and $h$ be two adjacent nodes in a Hamming graph and let $a$ be any positive real. Let us denote by $d_{ij}$ the distance between node $i$ and node $j$ in the graph.
I'm trying to find a ...

**6**

votes

**2**answers

410 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 $F_q$ ...

**1**

vote

**1**answer

331 views

### stabilizer always abelian? [closed]

Let G be a finite group acting on a finite dimensional vector space V.
Let C be a nontrivial subspace of V. Let H be the subgroup of G that
fixes C pointwise (the stabilizer of C). I'm fairly sure ...

**0**

votes

**1**answer

154 views

### Subset-Free Codes

For each non-negative integer $n$, what antichain(s) in $\{0,1\}^n$ with the pointwise partial order: $\;\;$ 1. $\;$ have the most elements $\;\;$ 2. $\;$ minimize the maximum of its elements' sum ...

**2**

votes

**1**answer

749 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 ...

**4**

votes

**1**answer

462 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 ...

**0**

votes

**1**answer

254 views

### Lovasz theta function - uses

Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?

**0**

votes

**1**answer

235 views

### Number of points on a complex sphere with pairwise inner product restriction

Considered the following inner products:
$(1)$ $\langle x,y \rangle = \sum_{t=1}^{n}x_{t}y_{t}$
$(2)$ $\langle x,y \rangle_{c} = \sum_{t=1}^{n}x_{t}\bar{y}_{t}$
consider the following surfaces:
...

**7**

votes

**2**answers

562 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
...

**9**

votes

**1**answer

294 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 ...

**5**

votes

**1**answer

1k 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 ...

**0**

votes

**0**answers

107 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 ...

**0**

votes

**1**answer

270 views

### Optimization problem related to parity check code

I have n data blocks and k parity blocks distributed across m boxes. Each parity block is Ex-or of some data block (for ease of understanding we can assume each data/parity block as a single bit) and ...

**2**

votes

**1**answer

311 views

### Lee codes and $n$-torus

This is in continuation with this post:
Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric
Codes over alphabet $\mathbb{Z}_{q}$ of length $n$ for the Lee metric ...

**1**

vote

**0**answers

223 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 ...

**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 ...

**2**

votes

**1**answer

272 views

### Doing column permutation under row overlap constraint

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 ...

**4**

votes

**1**answer

346 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 ...

**9**

votes

**5**answers

882 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 ...

**15**

votes

**1**answer

788 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 ...

**0**

votes

**1**answer

330 views

### Fixed Hamming distance property of binary deletion correcting codes

Let $x=(x_1x_2...x_n)$ be a binary sequence of length $n$. The Varshamov-Tenengolts code $VT_0(n)$ consists of all binary vectors $(x_1, . . . , x_n)$ satisfying $\Sigma_{i=1}^n i*x_i \equiv0 \pmod ...

**2**

votes

**1**answer

478 views

### Even lattices and binary codes

I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.
What is known about even ...

**14**

votes

**1**answer

992 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 $c_2$ ...

**1**

vote

**1**answer

286 views

### Large subgroups of the Hamming cube

Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube).
Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...

**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 ...

**1**

vote

**1**answer

434 views

### Matrix version of Berlekamp Massey algorithm

What are the most obvious generalizations of Berlekamp Massey algorithm [1] to matrix sequences?
[1] Massey, J. L., "Shift-register synthesis and BCH decoding", IEEE Trans. Information Theory ...

**2**

votes

**1**answer

281 views

### Optimal binary code for points in a metric+probability space

Does anyone know of any results on this topic?
Basically I'm considering this problem. You have some space $X$ from which you can draw points $x$ and $y$, a distance metric $d(x,y)$, and a ...

**4**

votes

**4**answers

626 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}),\\\
...

**10**

votes

**3**answers

617 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 ...

**5**

votes

**2**answers

410 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 ...

**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 ...

**1**

vote

**1**answer

330 views

### Algorithm for generating a size k error-correcting code on n bits

I want to generate a code on n bits for k different inputs that I want to classify. The main requirement of this code is the error-correcting criteria: that the minimum pairwise distance between any ...

**5**

votes

**2**answers

562 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 ...