Mod-2 Vector Arithmetic

1
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 ab 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 (a0.b0)(a1.b1)(an-1.bn-1)
• where all additions and multiplications are
  mod-2
• If dot product is zero, the vectors are
  orthogonal.

2
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 b1 and b2 are
  scalars from the set 0,1 and x1 and x2 are
  vectors from the set, then
• b1.(x1x2) (b1.x1) (b1.x2)
• (b1b2).x1 (b1x1) (b2.x1)
• An associative law is satisfied
• (b1.b2).x1 b1.(b2.x1)

3
Reference

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