The theory of error-correcting codes stems from Shannon's 1948 _A mathematical theory of communication_, and from Hamming's 1950 "Error detecting and error correcting codes".

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979 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 ...
14
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1answer
653 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
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1answer
838 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
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3answers
678 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
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3answers
569 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
1answer
271 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 ...
8
votes
2answers
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 ...
7
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3answers
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 ...
7
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4answers
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
3answers
577 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
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1answer
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
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2answers
540 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 ...
6
votes
5answers
515 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
3answers
310 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
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2answers
368 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
1answer
845 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
2answers
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 ...
5
votes
2answers
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 ...
5
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1answer
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 ...
5
votes
0answers
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 ...
5
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0answers
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 ...
4
votes
2answers
823 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
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3answers
262 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
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2answers
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 ...
4
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1answer
145 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
1answer
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
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4answers
605 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
2answers
339 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
1answer
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 ...
4
votes
1answer
181 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
1answer
397 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
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0answers
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 ...
3
votes
3answers
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 ...
3
votes
3answers
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?
3
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3answers
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
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1answer
145 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
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2answers
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}, ...
3
votes
1answer
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 ...
3
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1answer
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 ...
3
votes
2answers
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
2answers
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
2answers
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
1answer
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 ...
3
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1answer
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 ...
3
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1answer
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 ...
3
votes
2answers
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
0answers
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 $m$. $n(m)$ ...
2
votes
3answers
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 ...
2
votes
2answers
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 ...
2
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1answer
464 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 ...