The Hamming codes with n = 2 s - 1 , m = n - s , r = 3 are an example, invented by Marcel Golay in 1949 and Richard Hamming in 1950. Defining PM[s_] := IntegerDigits[Range[2 s - 1], 2, s] blocks of data of length m can be encoded with Join[data, Mod[data . Select[PM[s], Count[#, 1] > 1 &], 2]] while blocks of length n (and at most one error) can be decoded with Drop[(If[#  0, data, MapAt[1 - # &, data, #]] &)[ FromDigits[Mod[data .

Lehmer used a = 23 and m = 10 8 + 1 . … When m = 2 j - 1 is prime, however, even the rightmost digit repeats only with period m - 1 for many values of a . … For a register of size n the maximal period of 2 n -1 is obtained whenever x n + Apply[Plus, x taps - 1 ] is one of the EulerPhi[2 n - 1]/n primitive polynomials that appear in Factor[Cyclotomic[2 n - 1, x], Modulus  2] .

There is always at least some operator that satisfies the constraints of any given axiom system—though in a case like a  b it has k = 1 . Of the 274,499 axiom systems of the form {…  a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9} , while 964 allow only {1,7} . The only cases of 2 or less operators that appear with k = 2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}} .

Algebraic forms [for cellular automaton rules] The rules here can be expressed in algebraic terms (see page 869 ) as follows: • Rule 22: Mod[p + q + r + p q r, 2] • Rule 60: Mod[p + q, 2] • Rule 105: Mod[1 + p + q + r, 2] • Rule 129: Mod[1 + p + q + r + p q + q r + p r, 2] • Rule 150: Mod[p + q + r, 2] • Rule 225: Mod[1 + p + q + r + q r, 2] Note that rules 60, 105 and 150 are additive, like rule 90.

Assuming b > a > 0 , the number of zeros from the second family which appear between the n th and (n + 1) th zero from the first family is (Floor[(n + 1) #] - Floor[n #] &)[(b - a)/(a + b)] and as discussed on page 903 this sequence can be obtained by applying a sequence of substitution rules. For Sin[a x] + Sin[b x] a more complicated sequence of substitution rules yields the analogous sequence in which -1/2 is inserted in each Floor .

Array[Apply[Times, Map[(1 - Mod[#, 2])(# - 1)!!… But when n = 4 isotropy requires the {1, 1, 1, 1} and {1, 1, 2, 2} tensor components to have ratio β = 3 —while square symmetry allows these components to have any ratio. … And at least in the case of ordinary random walks, they do, so that for example, the ratio averaged over all possible walks of n = 4 tensor components after t steps on a square lattice is β = 3 + 2/(t - 1) , converging to the isotropic value 3, and the ratio of n = 6 components is 5 - 4/(t - 1) + 32/(3t - 4) .

Human generation of randomness If asked to type a random sequence of 0's and 1's, most people will at first produce a sequence with too many alternations between 0 and 1.

Note that the pattern shown here has been truncated at the edge of the page on the left, although in fact the whole pattern continues to expand to the left forever with an average slope of Log[2, 3]≃1.58 .

The particular rule used here can be described by the formula a i '= Mod[a i-1 + a i+1 , 2] .

Specifying an operator f (taken in general to have n arguments with k possible values) by giving the rule number u for f[p, q, …] , the rule number for an expression with variables vars can be obtained from With[{m = Length[vars]}, FromDigits[ Block[{f = Reverse[IntegerDigits[u, k, k n ]] 〚 FromDigits[ {##}, k] + 1 〛 &}, Apply[Function[Evaluate[vars], expr], Reverse[Array[IntegerDigits[# - 1, k, m] &, k m ]], {1}]], k]]