2
votes
0answers
40 views

Uniqueness of projective plane of order 5

Is there a slick way to see the uniqueness of projective plane (equivalently, an affine plane) of order $5$?
4
votes
0answers
91 views

Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
3
votes
1answer
75 views

Counting equivalent funtions over GF(2^n)

Let $F = GF(2^n)$ specified using a particular irreducible polynomial of degree $n$. Now, let $f(x) = x^{-1}$ in $F$, except we define $f(0):=0$. Let $G$ and $H$ be invertible $n \times n$ matrices ...
6
votes
0answers
48 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
1
vote
0answers
85 views

Number of ways to decompose the space $\mathbb F^n_2$ into a direct sum of two spaces

How many ways can $\mathbb F^n_2$ be decomposed into a direct sum of two subspaces? Basically how do I find the number of decompositions $\mathbb F^n_2 = \mathbb F^k_2 \bigoplus \mathbb F^{n-k}_2$ ...
0
votes
1answer
64 views

Finding maximal number of bad triplets

Let $a,b,c\in \mathbb{F}_{3^n}$. The summation of two vectors is done with modulo $3$. The elements of vectors are $0,1$ or $2$. We will say that $a,b,c$ form a bad triplet if $a\neq b,a\neq c,b\neq ...
0
votes
0answers
52 views

Condition for the number of distinct solutions over GF($q$)

Assume that we have $p$ sets $\left\{ {{m_i}} \right\}_{i = 1}^p$ with given cardinalities $\left\{ {{K_i}} \right\}_{i = 1}^p$, $1 \le {K_i} \le q$, where $q$ is a power of $2$. What I'm trying to do ...
15
votes
1answer
194 views

Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
3
votes
2answers
117 views

Geometric solution to classic committee problem

Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ...
5
votes
0answers
125 views

Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
2
votes
2answers
151 views

The linearity of the extended code

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is: If the code $C$ is linear, can we prove that the extended code $C'$ is linear too? ...
1
vote
1answer
82 views

extended linear codes over the field $\mathbb F_q$

Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where $$ C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ...
2
votes
1answer
747 views

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime. I'd like to start off by acknowledging that I know there are many posts relating to similar ...
4
votes
1answer
47 views

number of 1-to-1 linear functions on vectorspaces over finite fields

This is not a homework. I just ask this question myself and thought it would be easy to figure out. But I did not get the solution. Let $\mathbb{F}$ be a finite field with $|\mathbb{F}|=q$. Consider ...
2
votes
1answer
79 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
0
votes
1answer
61 views

Could you show me an example of an order 7 pandiagonal latin square?

Could you show me an example of an order 7 pandiagonal latin square? A pandiagonal latin square is one where no broken diagonal contains repeated symbols. I have found examples for smaller order but ...
11
votes
2answers
466 views

Whats the probability a subset of an $\mathbb F_2$ vector space is a spanning set?

Let $V$ be an $n$-dimensional $\mathbb F_2$ vector space. Note that $V$ has $2^n$ elements and $\mathcal P(V)$ has $2^{2^n}$. I'm interested in the probability (under a uniform distribution) that an ...
-1
votes
2answers
161 views

A pair of MOLS of order 15?

Is there any place where I can find a pair of MOLS(mutually orthogonal latin squares) of order 15? I can't seem to find a place where it's spelled out explicitly.
0
votes
1answer
145 views

Proof for latin squares- disjoint transversals

I cannot find a proof for the following theorem anywhere: A latin square has an orthogonal mate iff it can be decomposed into disjoint transversals. Could you perhaps link me to one? Also, how can ...
1
vote
1answer
349 views

The number of subspaces of a vector space

Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, and let $U$ be a subspace of dimension $k$. I want to compute the number of subspaces $W$ of $V$ of dimension $m$ such that $W\cap U=0$. ...
6
votes
2answers
192 views

Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?

Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet) Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n ...
2
votes
3answers
564 views

Product of all irreducibles with degree divisible by $n$ in $\mathbb{F}_{q^n}$?

In the finite field of $q^n$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to simply be $X^{q^n}-X$. Why is this? I understand that $q^n=\sum_{d\mid ...
5
votes
1answer
187 views

Generating function identity from number of irreducible monic polynomials in $\mathbf{GF}(q)$.

I denote by $m_n(q)$ the number of irreducible monic polynomials with degree $n$ over the finite field of order $q$. So the number of monic polynomials with degree is just $q^n$. From this, how does ...
12
votes
1answer
348 views

Squarefree polynomials over finite fields

I'm trying to figure out how many squarefree polynomials there are of a fixed degree over $\mathbb{F}_2$ specifically (and in general, over any finite field). Looking at some low-degree examples seems ...
2
votes
3answers
97 views

Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$

Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field. $\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane. Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$. I look at a line equation ...
6
votes
2answers
186 views

A resemblance between 2 binomial identities - why?

Let $F$ be any field (or even ring). The following formal power series identity (i.e., equality in $F[[x]]$) holds for any $j \ge 0$: $$(1-x)^{-j} = \sum_{i \ge 0} \binom{i +j -1}{i} x^i $$ The ...
10
votes
1answer
353 views

Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...