Yes, those are the lexicographically smallest. Were you asking for the ones with fewest terms? Something else? It wasn't clear to me. Just done with a brute-force search. (I wrote some code a while back when I was playing with finite fields for a demonstration of secret sharing. Ended up with a python library with classes for such things as finite fields, rationals/complex rationals/quaternion rationals, matrices, polynomials. (https://github.com/ms0/pymath) The question I asked at the start of this thread was really about short-circuiting that brute-force search a bit, at least for odd characteristic. I'm testing with the deterministic Rabin irreducibility test. Of course, I can easily generate a similar table for other primes. (Python isn't fast, but it's easy to code in; the p=2 case does everything with bitwise operations, so it's not as slow as it could be.) On 18-Jan-21 16:43, James Propp wrote:
Thanks, Mike!
Looks like you tackled all the Galois fields GF(2^n), not just GF(2^(2^k)). How did you do it?
Let me make sure I understand your notation. Does GF2_15_11 mean the polynomial 1 + x + x^15?
I am surprised that the bit strings you found are so short (corresponding to most of the high-degree coefficients being zero). It makes me wonder whether you answered a different question than the one I asked, and computed the lexicographically first irreducible polynomial of degree n. That’s an interesting question too, but I see no reason why they should have the same answer.
Jim Propp
On Mon, Jan 18, 2021 at 3:35 PM Mike Speciner <ms@alum.mit.edu <mailto:ms@alum.mit.edu>> wrote:
Here's the first 64, with the x^n term elided; constant term last. Not an obvious pattern, and, interpreted as a binary number, not in OEIS (either with or without the x^n term)...
'GF2_2_0' 'GF2_2_11' 'GF2_3_11' 'GF2_4_11' 'GF2_5_101' 'GF2_6_11' 'GF2_7_11' 'GF2_8_11011' 'GF2_9_11' 'GF2_10_1001' 'GF2_11_101' 'GF2_12_1001' 'GF2_13_11011' 'GF2_14_100001' 'GF2_15_11' 'GF2_16_101011' 'GF2_17_1001' 'GF2_18_1001' 'GF2_19_100111' 'GF2_20_1001' 'GF2_21_101' 'GF2_22_11' 'GF2_23_100001' 'GF2_24_11011' 'GF2_25_1001' 'GF2_26_11011' 'GF2_27_100111' 'GF2_28_11' 'GF2_29_101' 'GF2_30_11' 'GF2_31_1001' 'GF2_32_10001101' 'GF2_33_1001011' 'GF2_34_11011' 'GF2_35_101' 'GF2_36_110101' 'GF2_37_111111' 'GF2_38_1100011' 'GF2_39_10001' 'GF2_40_111001' 'GF2_41_1001' 'GF2_42_100111' 'GF2_43_1011001' 'GF2_44_100001' 'GF2_45_11011' 'GF2_46_11' 'GF2_47_100001' 'GF2_48_101101' 'GF2_49_1110001' 'GF2_50_11101' 'GF2_51_1001011' 'GF2_52_1001' 'GF2_53_1000111' 'GF2_54_1111101' 'GF2_55_1000111' 'GF2_56_10010101' 'GF2_57_10001' 'GF2_58_1100011' 'GF2_59_1111011' 'GF2_60_11' 'GF2_61_100111' 'GF2_62_1101001' 'GF2_63_11' 'GF2_64_11011'
On 18-Jan-21 12:01, James Propp wrote: > If I’m not mistaken, Conway’s notion of Nim-multiplication gives a way of > singling out a “first” degree n irreducible polynomial over GF(2). For > instance, x = *4 satisfies x^2 = *6, x^3 = *14, and x^4 = *5, so the first > irreducible polynomial of degree 4 is x^4 + x + 1. > > It would be interesting to peruse these polynomials for the first dozen > powers of 2 to see if there’s a simple pattern. > > When n isn’t a power of 2 you need to use Nim-values associated with > infinite ordinals but that’s okay since they’re well-ordered. > > Jim Propp > > On Mon, Jan 18, 2021 at 7:23 AM Joerg Arndt <arndt@jjj.de <mailto:arndt@jjj.de>> wrote: > >> Possibly making a fool of myself... >> Have you checked 'the Bible' (Lidl, Niederreiter)? >> I seem to recall that there is a nontrivial amount >> of coverage about binomial polynomials, specifically >> regarding irreducibility. >> >> Best regards, jj >> >> >> * Mike Speciner <ms@alum.mit.edu <mailto:ms@alum.mit.edu>> [Jan 14. 2021 17:11]: >>> Let $n = 2^{e_0} q_1^{e_1} \cdots q_i^{e_i}$ be the prime factorization >> of >>> $n$. >>> Let $p$ be an odd prime. >>> Is it true that >>> $\exists k$ such that $x^n+k$ is irreducible over GF($p$) $\iff$ >>> $(2^{min(e_0,2)} q_1 \cdots q_i) | (p-1)$ ? >>> >>> Are there similar criteria for "larger" polynomials such as $x^n + k_1 x >> + >>> k_0$ ? >>> >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun <https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun> >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun <https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun> >> > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun <https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>