**4**

votes

**2**answers

240 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 ...

**5**

votes

**0**answers

198 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 ...

**1**

vote

**1**answer

170 views

### Extended Hypercube Graph

Definition 1. The $n$-hypercube graph has vertices which are the elements of the set $\lbrace 0,1\rbrace^n$ of $n$-bit binary strings, and an edge is drawn between each pair of vertices representing a ...

**4**

votes

**3**answers

382 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 ...

**1**

vote

**2**answers

476 views

### Hamming codes from overlapping vectors

I am interested in whether the following problem is known.
For a given binary vector $V$ of length $n\geq m$, let $S$ be a subset of the possible subvectors of $V$ of length $m$ and say that the size ...

**0**

votes

**1**answer

162 views

### Isometry on a Hamming cube

Let $E^n$ be a Hamming cube of dimension $n$, and $\phi$ be a mapping from $E^n$ to $E^n$ that preserves Hamming distance, i.e. $d(x,y)=d(\phi (x),\phi (y))$. The question is the following: show that ...

**3**

votes

**0**answers

182 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 $m$.
$n(m)$ ...

**4**

votes

**1**answer

161 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 ...

**1**

vote

**1**answer

124 views

### (A)periodicity and (In)dependence on the boundary condition for some discrete analog of ODE (convolutional codes)

(See also MO117508, MO116611). This post describes somewhat real problem with convolutional codes. Let me first try to give brief and vague formulation of the question, later give details.
Problem ...

**3**

votes

**1**answer

244 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 ...

**4**

votes

**1**answer

278 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

**1**answer

235 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 ...

**7**

votes

**2**answers

533 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 ...

**4**

votes

**2**answers

274 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}, ...

**1**

vote

**1**answer

533 views

### How to find next to optimal path in hidden Markov model or what should be LIST-Viterbi algorithm ?

The Viterbi algorithm is an algorithm for finding the most likely sequence of hidden states – called the Viterbi path.
Question If I am interested in list of several paths - optimal, sub-optimal, ...

**0**

votes

**1**answer

124 views

### How many combinations exist of $M'$ items from a set of $M$ items such that each combination is not similar at more than $m$ elements?

I apologize if this has been answered before.
I would like to know how many ways there are to choose $M'$ elements from a set of $M$ elements such that any two sets selected are not similar at more ...

**0**

votes

**1**answer

316 views

### What are “best” polynoms f(x) g(x) of degree n ? I.e. ideal generated by them is as far from zero as possible ? (Best convolutional codes.)

Consider polynoms f(x) g(x) of degree at most n. (I am mostly interested about F_2[x]).
Let us multiply them by arbitrary polynoms p(x) i.e. consider ideal (p f , p g) in $F_2[x]\oplus F_2[x]$.
Let ...

**1**

vote

**1**answer

713 views

### “Trellis graph” is it standard term in graph theory ? What are its properties ?

In coding theory (convolutional codes) the graph called "trellis diagramm" is used
to visualize something.
I wonder is it a standard term in graph theory? Corresponding Wikipedia article is not
...

**2**

votes

**0**answers

128 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 ...

**0**

votes

**1**answer

185 views

### Multiplication by polynomials x^2+1 ; x^2+x+1. Does minimal Hamming norm of image equal to 5 ?

Everything over F_2. Let us define Hamming norm of polynom |p(x)| = number of non-zero monoms.
Respectivly for a pair of polynoms |[p ; g]| = |p| +|g|.
Consider linear map $F_2[x] \to F_2[x] \oplus ...

**3**

votes

**3**answers

549 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 ...

**1**

vote

**1**answer

200 views

### Given g1(x), g2(x) minimize over p(x) Hamming weight of [p(x)g1; p(x)g2(x) ] ? (Or how to find minimal distance of convolutional code?)

Fix polynoms g1(x), g2(x) over F_2[x].
Question: How to find minimum over polynoms p(x) of the:
HammingWeight(p(x) g1(x) ) + HammingWeight(p(x) g2(x) ) ?
By HammingWeight of polynom I mean number ...

**2**

votes

**3**answers

385 views

### A different criterion for equivalence of codes?

I've been thinking about equivalence of codes (two codes that are equal up to order of positions of the letters, or permutations of the letters in a fixed position).
It is obvious that if we have two ...

**3**

votes

**3**answers

449 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

**1**answer

255 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 ...

**2**

votes

**1**answer

346 views

### Error correcting codes obtained as superposition of two codes e.g. CRC+Convolutional

Setup reminder: linear block error-correcting code is some linear subspace $C$ in $F_2^N$. (Correcting error means to find a point $c \in C$ which is "nearest" to a given $r$ in $F_2^N$, $r$ is ...

**3**

votes

**1**answer

323 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 ...

**2**

votes

**1**answer

260 views

### What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

Everything over F_2.
For any matrix $A$ define the number $N(A) = min_{x}$ HammingWeight $( [x , Ax])$.
Where $x$ is vector and [a,b] is just concatenation of vectors: (a_1,...a_n, b_1,...,b_m).
...

**1**

vote

**1**answer

191 views

### How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Let us consider polynoms over $F_2$.
Consider the linear SUBSPACE of polynoms divisible by $x^{16}+x^{12}+x^5 +1$ and of degree less or equal $N$ (e.g. 40).
Question: How many k-nomials belong to ...

**2**

votes

**1**answer

208 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 ...

**0**

votes

**1**answer

318 views

### Is there any relationship between a tree(graph theory) and semi-metric?

suppose we have a tree(undirected) with $n$ vertices.The edges are weighted(distances). Is it possible to impose a semi-metric structure on the graph using these distances and adjacency matrix?

**3**

votes

**2**answers

178 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 ...

**1**

vote

**1**answer

435 views

### A covering problem for the Hamming cube

Consider the set of all $k$-subsets of $\{1,\dots,n\}$, naturally identified with a subset $A$ of $\{0,1\}^n$ where each element has exactly $k$ ones. Is there a sharp bound known for ...

**13**

votes

**3**answers

796 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 ...

**1**

vote

**1**answer

235 views

### Will “error locating codes” have higher rates than ECCs?

I'm wondering to detect all the errors (i.e. their positions) in a codeword $(c_0, c_1, \cdots, c_{n-1})\in Q$ where $Q$ is an alphabet set with size $q$, i.e., to know whether $c_i$ is faulty or not, ...

**9**

votes

**0**answers

320 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

**1**answer

194 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.
...

**5**

votes

**2**answers

1k 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 ...

**3**

votes

**3**answers

2k 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 ...

**8**

votes

**3**answers

998 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 ...

**6**

votes

**2**answers

1k 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 ...

**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} ...

**1**

vote

**1**answer

960 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

284 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 ...

**4**

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

330 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

333 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

635 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

165 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

473 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 ...