# Tagged Questions

error-correcting codes, error-detecting codes and related algebraic and/or combinatorial constructions

19 views

18 views

42 views

### Over $F_5$, why does $f(1)=-2$ where $f(x)=x^2+2$

I am working over $F_5$ and $f(x)=x^2+1$ I am told that $f(1)=-2$. I understand that $-2=3$mod$5$ Why can we not leave it as $f(1)=1^2+2=3$? Because $3$ mod $5$ $=3$ so why do we have to "change" ...
40 views

### Is the polynomial $x^2 + x+2$ primitive?

Let $F_5$ be the field of integers modulo $5$. I am trying to find out if $x^2 + x + 2$ is primitive or not. So first we see that the divisors of $24$ are $1,2,3,4,6,8,12$. We find $$x^2=-x-2$$ ...
15 views

### Does this definition of a primitive polynomial make sense?

I have this question in a past exam paper. Let $F_p$ be the field of integers modulo prime $p$. I have the question What is meant by saying $f$ is primitive? This is the solution I have. ...
17 views

### How to decompose a Finite State Machine (FSM) into a cascade of two or more FSMs?

I am coming from Electrical Engineering background and would like to know how can I decompose a given FSM into a cascade of two or more FSMs. To be more precise, I am looking at following questions ...
27 views

### Permutations acting on coordinates of codewords

Let $\mathcal{C}$ be a binary code of length $n$. The automorphism group of $\mathcal{C}$ is defined to be the set of permutations in $S_n$ that take $\mathcal{C}$ to itself. The text by MacWilliams ...
42 views

### Finding modular inverse of every number mod 26?

I am looking at cryptography, and need to find the inverse of every possible number mod 26. Is there a fast way of this, or am i headed to the algorithm every time?
39 views

### Proof regarding size and dimension of linear codes

The problem is stated as follows: Let C be a binary linear code of length n, dimension k and distance d and assume that C contains at least one element of odd weight. Let C' be the subset of C ...
36 views

### Repetition code and binary symmetric channel, where error is near 1/2

I want to send one bit $x$ over a noisy channel, specifically, a binary symmetric channel with error probability $p$, where $p=(1-\epsilon)/2$ and $\epsilon$ is small. In other words, the error ...
12 views

33 views

14 views

### Channel capacity of sum of symmetric channels

I've got a channel matrix $P$ of the form $\begin{bmatrix} Q \\ R \end{bmatrix}$ where $Q,R$ are channel matrices of symmetric channels, so they now have different input alphabets but the ...
17 views

### Dimension of repetition code

I read a proof that the dimension of a repetition code is 1 using the equivalence of $Rep_{n,k}$ with $Span\{e_1\}$ but it's not very intuitive for me and I'd like to understand it using the ...
63 views

### Decode the message $(1,1,1,0,1,1,1)$ using the Hamming $(7,4)$ code

The question is asking me to decode $(1,1,1,0,1,1,1)$ using Hamming $(7,4)$ code. I know that I am suppose to set a $3 \times 7$ matrix ${\bf H}$ and multiply it by ${\bf r}$ and set it equal to zero, ...
18 views

### Extended Golay codes are self dual

Show that extended Golay code $G_{24}$ and $G_{12}$ are self dual. To show it have to show that any two rows of $G_{12}$ and $G_{24}$ are orthogonal, that is inner product of any two rows are zero. ...
13 views

### Relation between Completely regular codes and Perfect codes

I have a question about completely regular codes. I know that each perfect code is a completely regular codes? Is it right? If it is right, I want to know the proof.
Suppose that we have a binary cyclic code $C$ of length $n \geq 3$ with generator polynomial $b(x) \neq 1$, where $n$ is the smallest natural number such that $b(X) \mid X^n-1$. I want to show that ...