# Tagged Questions

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### 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 ...

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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 ...

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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**

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**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 ...

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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 ...

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**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 ...

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211 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 ...

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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 ...

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**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 ...

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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 ...

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**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 – called the Viterbi path.
Question If I am interested in list of several paths - optimal, sub-optimal, ...

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**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 ...

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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 ...

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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 ...

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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 ...

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**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 ...

**3**

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**3**answers

400 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?

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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 ...

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201 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).
...

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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 ...

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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 ...

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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?

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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 ...

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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, ...

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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 ...

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748 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 ...

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744 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 ...

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729 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 ...

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222 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 ...

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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 ...

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281 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 ...

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265 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 ...

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**1**answer

148 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 ...

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367 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 $F_q$ ...

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225 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:
...

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201 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 ...

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**1**answer

242 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 ...

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233 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 ...

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271 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 ...

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527 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 ...