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 Apr11 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). Apr11 answered “Consecutive” square residues in odd-order finite fields Apr10 asked “Consecutive” square residues in odd-order finite fields Apr7 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? Apr7 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. Apr7 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 Apr6 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 Apr6 asked Orthogonal matrices over GF(2^t) with the first column fixed Mar13 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 $\oplus$ Mar13 revised Cayley-Dickson construction: a general rule for multiplying imaginary units? Corrected the formula for $N_{x,y}$ Mar11 revised Cayley-Dickson construction: a general rule for multiplying imaginary units? Updated in light of my answer Mar11 answered Cayley-Dickson construction: a general rule for multiplying imaginary units? Mar6 asked Cayley-Dickson construction: a general rule for multiplying imaginary units? Mar5 revised A puzzle on orthogonal matrices modulo $p$ Removed reference to higher dimensional examples after discovery of some complications Mar5 awarded Excavator Mar5 revised A puzzle on orthogonal matrices modulo $p$ Major revision in presentation, increased the generality of the statement Mar5 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. Mar5 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? Mar4 revised When (and how often) is $2^k+1$ a prime power? Edited to make the answer self-contained Mar4 accepted When (and how often) is $2^k+1$ a prime power?