Reputation
4,180
Top tag
Next privilege 5,000 Rep.
Approve tag wiki edits
Badges
10 30
Newest
 Excavator
Impact
~86k people reached

Apr
11
comment “Consecutive” square residues in odd-order finite fields
@JyrkiLahtonen: Thanks! I found a yet simpler, turn-the-crank sort of proof once it occurred to me that the answer might possibly be a straightforward extension of the standard result (and that the standard result might not only be well-known but might also itself be proven simply).
Apr
11
answered “Consecutive” square residues in odd-order finite fields
Apr
10
asked “Consecutive” square residues in odd-order finite fields
Apr
7
comment Orthogonal matrices over GF(2^t) with the first column fixed
@JyrkiLahtonen: So, my original motivation is in fact to study orthogonal matrices over GR(q,q^t) for q a power of 2... simplifying to GF(2^t) might have opened up new solutions which do not easily Hensel lift. Do you think it might be best for me to roll back my question?
Apr
7
comment Orthogonal matrices over GF(2^t) with the first column fixed
@JyrkiLahtonen: Thank you. I think that one might need at least 4x4 blocks because of the case of vectors $\mathbf v$ containing only 1s and 0s, to produce 1s in pairs. (I'm using a similar technique to deal with "unitary" matrices over Galois Rings which quadratically extend other Galois Rings.) This same trick may indeed work for GF(2^t), but sadly it is not clear that it would Hensel lift to orthogonal matrices over GR(q,q^t) for q a power of 2, unless it so happened that all of the other columns were unit vectors mod 4 as well as mod 2.
Apr
7
revised Orthogonal matrices over GF(2^t) with the first column fixed
Minor revisions to improve presentation, as well as to remove redundant information left over from first version
Apr
6
revised Orthogonal matrices over GF(2^t) with the first column fixed
Focused on a special case likely to give rise to an answer to the more general question
Apr
6
asked Orthogonal matrices over GF(2^t) with the first column fixed
Mar
13
revised Cayley-Dickson construction: a general rule for multiplying imaginary units?
Revised description (the formula is no longer "closed form" in presentation), removed redundant description of $\oplus$
Mar
13
revised Cayley-Dickson construction: a general rule for multiplying imaginary units?
Corrected the formula for $N_{x,y}$
Mar
11
revised Cayley-Dickson construction: a general rule for multiplying imaginary units?
Updated in light of my answer
Mar
11
answered Cayley-Dickson construction: a general rule for multiplying imaginary units?
Mar
6
asked Cayley-Dickson construction: a general rule for multiplying imaginary units?
Mar
5
revised A puzzle on orthogonal matrices modulo $p$
Removed reference to higher dimensional examples after discovery of some complications
Mar
5
awarded  Excavator
Mar
5
revised A puzzle on orthogonal matrices modulo $p$
Major revision in presentation, increased the generality of the statement
Mar
5
comment A puzzle on orthogonal matrices modulo $p$
After a short depth-first search on the relevant concepts and terminology, it would appear that these are examples of orthogonal designs. I hope you don't mind if I edit your answer to present the idea in more generality.
Mar
5
comment A puzzle on orthogonal matrices modulo $p$
It is clear that this construction generalizes to matrices of any order $2^n$, using the Cayley-Dickson construction: for example, the $8\times8$ case would use the octonians, the $16\times16$ case would use the sedenions, etc. Thus for any "unit" vector (modulo any $k$) of size $2^n$, one may obtain an orthogonal matrix (modulo $k$) having that vector as its first column. Does this construction have a name?
Mar
4
revised When (and how often) is $2^k+1$ a prime power?
Edited to make the answer self-contained
Mar
4
accepted When (and how often) is $2^k+1$ a prime power?