Things have progressed again and a new branch of the PEAC family has appeared, tuned for SW implementations where the masking operation is simply skipped. This completely changes the mathematics of the system.

Below are 2 examples with w26 and w31 which are "extended" (hence the "+") by not masking the sum after the addition.

w26 is known to be "perfect" but w31 does not pass the sieve.

However messing with the MSB(s) changes the rules and everything has to be reconsidered now. The difficulty is that the 32-bit words require immense amounts of computations and the others (based on 64-bit registers) are even worse.

So a thorough computation exploration is required for

• w26+
• w31 (not holding my breath)
• w31+
• w63 (passes sieve)
• w63+
• w64

At least a first estimate of computability of the primary orbit.

Furthermore, another class of gPEAC needs to be evaluated: the 1.5× types, called "pseudo-binary", for use in Line Scramblers (see the previous logs such as 167. Line encoding with PEAC: it's alive) : the 2 MSB of the modulo are 1 and the rest is 0. These are powers of 2 multiplied by 3, so the subtraction only requires processing a few MSB and not the whole word. At least a few of these numbers should be in the list of the moduli I had scanned last year. I have found 3 only...

a=3 ; for i in $(seq 40); do a=$(($a*2)); echo $a; done
6 : perfect
12 : 10P
24 : maximal
48 : maximal
96 : maximal
192 : 191M 186P
384 : 380M, 378P, 374P
768 : 763M, 761M, 754P, 750P, 746P (w8)
1536 : 1528M, 1526P, 1516M, 1511M, 1504M, 1501M, 1498P
3072 : 3060M, 3038P, 3036M, 3018P
6144 : 6128M, 6118P, 6106P
12288 : 12280M, 12254P
24576 : 24563M, 24551M, 24549M, 24546P
49152 : 49050, 49084, 49094, 49128
98304  : 98300, 98298, 98285 98264 98254
196608 : 196605M 196598P 196594P          (w16)
393216 : 393208M 393195M 393186P 
786432 : 786418P  786403M 786388M 786390P 786391M
1572864
3145728
6291456
12582912
25165824
50331648
100663296
201326592
402653184
805306368
1610612736
3221225472
6442450944
12884901888
25769803776
51539607552
103079215104
206158430208
412316860416
824633720832
1649267441664
3298534883328
...

...

Stay tuned.