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Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M₂, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) ..., Prove that: - the length of C3 is 2n the number of codewords is M₁M2 the distance in otly d

Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M₂, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) ..., Prove that: - the length of C3 is 2n the number of codewords is M₁M2 the distance in otly d

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 36E
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Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M2, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) Prove that: - - the length of C3 is 2n the number of codewords is M₁M₂ the distance is exactly d3 Hence: C3 is a binary (2n, M₁M2, d3)-code
Transcribed Image Text:Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M2, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) Prove that: - - the length of C3 is 2n the number of codewords is M₁M₂ the distance is exactly d3 Hence: C3 is a binary (2n, M₁M2, d3)-code
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