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Mod-2 Vector Arithmetic

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

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Mod-2 Vector Arithmetic

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  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 = 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 = (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.(x1+x2) = (b1.x1) + (b1.x2) (b1+b2).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.

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