**1**

vote

**1**answer

45 views

### Function field Towers of larger depth of recursion

A function field tower is a sequence of function fields
$$\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \dots \subset \mathcal{F}_{n} \subset \mathcal{F}_{n+1} \subset \dots $$
over a base ...

**6**

votes

**3**answers

309 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 & ...

**2**

votes

**0**answers

67 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 ...

**1**

vote

**2**answers

100 views

### Designing almost orthogonal vectors in a deterministic manner

Consider the vector space $\mathbb{R}^n$, the standard inner product $\langle \cdot,\cdot \rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}$, and some $0<\epsilon\leq ...

**4**

votes

**0**answers

84 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 ...

**0**

votes

**0**answers

135 views

### Isomorphic Algebraic Geometric codes

Let $\chi/\Bbb F_q$ be an algebraic curve over a finite field $\Bbb F_q$ with $q^s$ rational points for some $s\in(0,1)$. Let $L(D)$ be the riemann roch space with prescribed zeros and poles. Let ...

**1**

vote

**1**answer

94 views

### On MDS code property

Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...

**1**

vote

**1**answer

266 views

### Cyclic Hamming Code

I know that Hamming codes can be arranged in cyclic form. But my question is how can I prove this.
My idea was to find a generator/primitive polynomial $p(x)$?
For example I want to show that the ...

**3**

votes

**2**answers

155 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

123 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

143 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?

**2**

votes

**0**answers

100 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 ...

**3**

votes

**1**answer

112 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 ...

**0**

votes

**1**answer

166 views

### Comparison Huffman Encoding and Arithmetic Coding dependent on Entropy

Where can I get an understanding of how Arithmetic Coding and Huffman Encoding compare as entropy increases. I know Arithmetic Coding is better for low entropy distributions, but how can I get a sense ...

**4**

votes

**2**answers

210 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

152 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

137 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

261 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

317 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

137 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

160 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

144 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

106 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

237 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

187 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

215 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 ...

**8**

votes

**2**answers

518 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 ...

**3**

votes

**2**answers

257 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

255 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 â€“ called the Viterbi path.
Question If I am interested in list of several paths - optimal, sub-optimal, ...

**0**

votes

**1**answer

118 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

249 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

465 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

120 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

173 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

463 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

188 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

335 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

399 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

233 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

260 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

240 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 ...

**1**

vote

**1**answer

200 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

184 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

**0**answers

156 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

273 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

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 ...

**1**

vote

**1**answer

326 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

676 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 ...

**0**

votes

**1**answer

222 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, ...

**5**

votes

**0**answers

243 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 ...