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**Closure**: For all*a*,*b*in A_{l}the result of*a*+*b*is also in A_{l}.**Associativity**: For all*a*,*b*,*c*in A_{l}the equation (*a*+*b*) +*c*=*a*+ (*b*+*c*) holds.**Identity element**: For the zero vector (denoted by '0') the equation 0 +*a*=*a*+ 0 =*a*for each value*a*in A_{l}.**Inverse element**: For all*a*in A_{l}the equation*a*+*a*= 0 holds, meaning that*a*is its own inverse element.**Commutativity**: For all*a*,*b*in A_{l}the equation*a*+*b*=*b*+*a*holds.

A product operator for binary vectors can be defined with a rotation
operator and the exclusive-or operator. The rotation operator **rot**_{i}
moves the bits in the vector *i* positions to the right, where elements
are wrapped around. The product of binary vectors *a* and *b* is defined
by adding all **rot**_{i}(*b*) vectors for which the *i*-th value
in vector *a* is equal to 1. When the product operator is represented by
the '·' sign and when the one vector, the vector where only the first
value is equal to 1 and all remaining value equal to 0, is represented
by '1', the the following hold:

**Closure**: For all*a*,*b*in A_{l}the result of*a*·*b*is also in A_{l}.**Associativity**: For all*a*,*b*,*c*in A_{l}the equation (*a*·*b*) ·*c*=*a*· (*b*·*c*) holds.**Identity element**: For the one vector the equation 1 ·*a*=*a*· 1 =*a*for each value*a*in A_{l}.**Commutativity**: For all*a*,*b*in A_{l}the equation*a*·*b*=*b*·*a*holds.

*a*· (*b*+*c*) = (*a*·*b*) + (*a*·*c*)- (
*a*+*b*) ·*c*= (*a*·*c*) + (*b*·*c*)

**rot**_{i}·**rot**_{j}=**rot**_{(i+j) modulo l}- For all
*a*,*b*in A_{l}:**rot**_{i}(a) +**rot**_{i}(b) =**rot**_{i}(a + b) - For all
*a*,*b*in A_{l}:**rot**_{i}(a) · b =**rot**_{i}(a · b)

**reorder**_{i}·**reorder**_{j}=**reorder**_{j}·**reorder**_{i}=**reorder**_{i · j modulo l}.**reorder**_{i}·**rot**_{j}=**rot**_{k}·**reorder**_{i}where*k*=*i*·*j*^{-1}**modulo***l*if*j*^{-1}, the inverse of*j*, defined by*j*·*j*^{-1}**modulo***l*= 1, exists.

1 1 2 2 10 4 3 100 8 4 1100 8 5 11000 16 5 11110 16 6 101000 16 6 111000 16 6 111100 16 7 1110100 8 8 11110000 32 9 100100000 64 9 110110000 64 9 111111000 64 10 1111100000 64 10 1011101000 64 12 101111010000 32 14 11011110010000 32 15 111101011001000 16My gut feelings tells me that there is a pattern of length 31 which can only reach 32 different states and that all patterns with lengths in between reach 32 or more states. If you search for the sequence 111101011001000 on the internet it will turn up some interesting results. It appears that the sequence can be produced by a four state shift register and exclusive-or operator. In this sequence each digit is the exclusive-or of the third and fourth digit to the right.

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