Mod-2 Vector Arithmetic For 2 binary n-vectors a and b –All components of a and b are elements of {0,1} –The mod-2 sum, c = a+b is term-by-term mod-2 sum of the vector components –The scalar product, c = ba of vector a and scalar b is the term-by-term mod-2 product of the vector components and scalar b –The dot product c = a.b is a scalar defined by a.b = (a 0.b 0 )+(a 1.b 1 )+……(a n-1.b n-1 ) where all additions and multiplications are mod-2 If dot product is zero, the vectors are orthogonal.

Binary Linear Vector Space Is a set of K binary n-vectors that satisfy: –Mod-2 sum of any 2 vectors in the set is another vector in the set –The mod-2 scalar product of an element of {0,1} and any vector in the set is another vector in the set –A distributive law is satisfied. If b 1 and b 2 are scalars from the set {0,1} and x 1 and x 2 are vectors from the set, then b 1.(x 1 +x 2 ) = (b 1.x 1 ) + (b 1.x 2 ) (b 1 +b 2 ).x 1 = (b 1 x 1 ) + (b 2.x 1 ) –An associative law is satisfied: (b 1.b 2 ).x 1 = b 1.(b 2.x 1 )

Reference R. E. Ziemer and R. L. Peterson, ‘Introduction to Digital Communications, 2 nd. Edition,’ Prentice Hall, 2001.