**12**

votes

**1**answer

581 views

### Maximum number of vectors in a hypercube satisfying given conditions

$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...

**2**

votes

**2**answers

261 views

### Minimally intersecting subsets of fixed size

The question I have, is how to generate the following collection of subsets:
Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...

**0**

votes

**1**answer

53 views

### Optimal covering and CSPNG

Consider a function $f: \{0,1\}^n \to \{0,1\}^{cn}$, where $c>1$.
A random $f$ with high probability generates optimal covering of $\{0,1\}^{cn}$,
i. e.:
$\forall x \in \{0,1\}^{cn}$ $\exists y ...

**0**

votes

**0**answers

37 views

### capacity of a channel with random phase rotation

Consider a wireless channel $h=e^{j\theta}$, where $\theta$ is a uniform random variable in $[0,2\pi]$ independent of the input messages and the independent of the noise. The channel randomly rotates ...

**0**

votes

**0**answers

22 views

### transform a matrix into block circulant form

What's known about transforming a matrix into block circulant form?
Given an $K \times N$ matrix $M$ (over GF(2) for now although the case
over ${\mathbb C}$ or ${\mathbb R}$ are also of interest), ...

**3**

votes

**1**answer

149 views

### Parity-check matrix for code with variable block size and minimum distance

Consider a linear error-correcting code with symbols in $GF(q)$, with codewords of length $k$ generated from messages of length $n$ and minimum distance $d+1 = k-n+1$. In the cases of interest, $q = ...

**1**

vote

**0**answers

96 views

### How can I decode efficiently a triple-error-correcting binary BCH code?

In a given $\mathrm{BCH}(N,K)$, $T=3$ code over $\mathrm{GF}(2^m)$, there are ways to find the error locations in a given $N$-bit codeword directly from the syndromes without going through the normal ...

**6**

votes

**0**answers

70 views

### irregular LDPC code construction algorithm

I want to construct a sparse random binary matrix ${{\bf{H}}_{m \times n}}$ that has the following properties
1- Faction of columns of weight $i$ is ${v_i}$ .
2- Fraction of rows of weight ...

**0**

votes

**2**answers

68 views

### Binary Codes and its distance [closed]

Find n binary codes of length n such that distance between each pair is n/2 , where n is a even number , if possible?How to generate all codes? for example if n=4 we have 1110,1101,1011,0111 each pair ...

**13**

votes

**3**answers

1k views

### A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...

**3**

votes

**1**answer

230 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 ...

**3**

votes

**1**answer

86 views

### Effects of many degree-2 variable nodes in the Tanner graph during the decoding of LDPC codes

Suppose that we have a LDPC code $C$ with a $(n -k )\times n $ parity check matrix $H$, and there exist approximately $ \sqrt n$ numbers of degree-2 columns. It means that there are approximately ...

**2**

votes

**2**answers

278 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, ...

**3**

votes

**0**answers

92 views

### Rank of a particular matrix

Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$.
Split variable set into ...

**4**

votes

**0**answers

111 views

### Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...

**3**

votes

**3**answers

3k 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 ...

**2**

votes

**0**answers

76 views

### The significance of the Parvaresh-Vardy curve

Even though this seems like a computer science question, it is purely mathematical and concerns polynomials and curves over finite fields.
Consider the Parvaresh-Vardy list decoder.
As I understand ...

**2**

votes

**1**answer

111 views

### Probability of Hamming weight

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v_i\}_{i=1}^{{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
Denote $v_j\cap v_j$ to be ...

**2**

votes

**0**answers

219 views

### Hamming weight probability of projections

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick $2^{n^t}$ random vectors $\{v_i\}_{i=1}^{2^{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$.
If $v_i^\perp$ is ...

**3**

votes

**1**answer

111 views

### Maximal neighbour-full partition of $\{0,1\}^n$

What is the largest complete minor of the $n$-dimensional hypercube? (which we call $k(n)$)
Alternatively, what is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that ...

**0**

votes

**1**answer

92 views

### Information theoretic common sequence agreement (not secret key)

Supposing Alice and Bob share $\rho$-correlated sequences in $\{0,1\}^n$, what coding theory based schemes are available for Alice and Bod to extract sequences $A,B\in\{0,1\}^n$ respectively such that ...

**9**

votes

**5**answers

839 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 ...

**1**

vote

**1**answer

108 views

### Coding for channels with concentrated error

Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$?
We know that if a channel with error flips (in the ...

**1**

vote

**0**answers

61 views

### How do you use the bits you get back from Bits Back Coding?

Bits Back coding is a scheme to transmit an observation x.
You can read about it here [1]. To my understanding, it works like this:
The encoder samples a message z from a distribution Q(z|x) that it ...

**5**

votes

**1**answer

140 views

### lower bound on A(k,4,floor(k/2))

A(k,4,r) is the independence number of the Johnson graph J(k,r).
What is the best known asymptotic lower bound on A(k,4,floor(k/2)) ?
I only obtained ...

**2**

votes

**2**answers

267 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 ...

**2**

votes

**2**answers

160 views

### Hamming graph and independent sets

I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) ...

**11**

votes

**2**answers

469 views

### “The Two Sheriffs” puzzle -2: threshold for security

I've already asked a question “The Two Sheriffs” puzzle with wrong assumption. Yoav Kallus in his amazing answer using Fano plane showed that the problem has a solution in the case of seven suspects.
...

**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 ...

**28**

votes

**1**answer

1k views

### “The Two Sheriffs” puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler.
Two sheriffs in neighboring towns are on the track of a killer, in a
case involving eight ...

**1**

vote

**1**answer

55 views

### Bound on the weight of the minimum weight generator of [n,k] cyclic codes?

I'm looking at creating sparse generator matrices for cyclic codes of a given length and dimension. A generator matrix of an [n,k] cyclic code can be expressed as
$G = \begin{bmatrix}g_0 & g_1 ...

**1**

vote

**1**answer

88 views

### Optimal covering

Let consider a problem of optimal covering of Hamming space.
So we have Hamming space $\{0,1\}^n$ and some integer $r$. We want to find a set $A \subseteq \{0,1\}^n$ such that any point from ...

**2**

votes

**0**answers

165 views

### PRNG and coding theory

Let $k, n \in \mathbb{N}$, $k = (1 - \epsilon)n$ where $1 >\epsilon > 0$.
I want to find $f: \{0,1\}^k \to \{0, 1\}^n$
such that:
1) $f(a) \not= f(b)$ if $a \not=b $
2) for any $x \in ...

**3**

votes

**2**answers

215 views

### Picking codewords that are close

I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...

**3**

votes

**3**answers

99 views

### Block error-correcting codes over inhomogeneous alphabets

For $n := (n_1,\dots,n_N) \in \mathbb{N}_{>1}^N$, let $X_n := \prod_{j=1}^N [n_j]$, where as usual $[m] := \{1,\dots,m\}$.
Are there any known generic constructions for (Hamming) sphere ...

**0**

votes

**0**answers

115 views

### Minimum-size spanning group using hamming distance

For given natural numbers $N$ and $D$, we will define $F$ as all the non-negative integers less than $2^N$. A group $A \subseteq F$ will be "spanning", if and only if every number $x$ in $F$ satisfies ...

**1**

vote

**1**answer

185 views

### Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...

**1**

vote

**0**answers

187 views

### Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote ...

**2**

votes

**0**answers

88 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

**0**

votes

**2**answers

394 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 ...

**3**

votes

**1**answer

125 views

### Balanced binary code that “resists” local decoding?

I am looking for a construction that can be stated as the following coding problem: a binary code with good distance ($d = \Omega(n)$ where codeword length is $n$) that "resists local decoding" in the ...

**33**

votes

**1**answer

576 views

### What measurable quantity can constrain the number of odors human can discriminate?

This is not a very typical MO question, but I hope you bear with me. It concerns a recent disagreement in the biology literature about how many different odors humans can discriminate. The authors of ...

**0**

votes

**2**answers

123 views

### Form of the Shannon capacity for Heptagon?

Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?

**9**

votes

**0**answers

324 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 ...

**-1**

votes

**1**answer

107 views

### Noise reduction in capacity-0 channels

Suppose we have a binary symmetric channel with $p=\frac{1}{3}$; that is, a communications channel in which each bit is flipped with independent probability $\frac{1}{3}$. I know that there is a code ...

**3**

votes

**0**answers

157 views

### On a problem of sphere-packing for Reed-Solomon codes

Suppose we have an $[n, k+1, n-k]$ Reed Solomon code $\mathcal C$ over $\mathbb F_q$, where $n-k=d$ is the minimum distance, and suppose that $d=2t+1$. We know that for every $r \in \mathbb F_q^n$ the ...

**3**

votes

**0**answers

116 views

### Applications of list decoding

This is citation from http://en.wikipedia.org/wiki/List_decoding:
Algorithms developed for list decoding of several interesting code families have found interesting applications in computational ...

**2**

votes

**1**answer

128 views

### Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution.
The mutual information between the input and the output of ...

**2**

votes

**1**answer

270 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).
...

**3**

votes

**1**answer

816 views

### “Codes” in which a group of words are pairwise different at a certain position

I read the following problem, claimed to be in the IMO shortlist in 1988:
A test consists of four multiple choice problems, each with three options, and the students should give an unique answer ...