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

The value of the cell at position n from the end of row t is thus the n th digit of m t , or Mod[Quotient[m t , k n ], k] .

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 .

LFSR cryptanalysis Given a sequence obtained from a length n LFSR (see page 975 ) Nest[Mod[Append[#, Take[#, -n] . vec], 2] &, list, t] the vector of taps vec can be deduced from LinearSolve[Table[Take[seq, {i, i + n - 1}], {i, n}], Take[seq, {n + 1, 2n}], Modulus  2] (An iterative algorithm in n taking about n 2 rather than n 3 steps was given by Elwyn Berlekamp and James Massey in 1968.)

The arithmetic system takes the value n that it obtains at each step, computes Mod[n, 30] , and then depending on the result applies to n one of the arithmetic operations specified by the rule above.

The pictures on the right below show Sin[1/2 π a[t, n]] 2 for these functions (equivalent to Mod[a[t, n], 2] for integer a[t, n] ).

Cyclic addition After t steps, the dot will be at position Mod[m t, n] where n is the total number of positions, and m is the number of positions moved at each step.

The number with run lengths corresponding to successive integers (so that the n th digit is Mod[Floor[1/2 + Sqrt[2n]], 2] ) turns out to be (1 - 2 1/4 EllipticTheta[2, 0, 1/2] + EllipticTheta[3, 0, 1/2])/2 , and appears at least not to be algebraic.

Equivalential calculus Expressions with variables vars are equivalent if they give the same results for Mod[Map[Count[expr, #, {-1}] &, vars], 2] With n variables, there are thus 2 n equivalence classes of expressions (compared to 2 2 n for ordinary logic).

Second-order cellular automata Second-order elementary rules can be implemented using CA2EvolveList[rule_List, {a_List, b_List}, t_Integer] := Map[First, NestList[CA2Step[rule, #]&, {a, b}, t]] CA2Step[rule_List, {a_, b_}] := {b, Mod[a + rule 〚 8 - (RotateLeft[b] + 2 (b + 2 RotateRight[b])) 〛 , 2]} where rule is obtained from the rule number using IntegerDigits[n, 2, 8] .